Displaying 1-10 of 22 results found.
Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!*1!*2!*...*n!).
+10
37
1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000
COMMENTS
Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001 a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009 Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
FORMULA
a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001. a(n) ~ A * exp(n^2/4 + n + 1/6) * n^(n^2/2 + 7/12) / (2^((n+1)^2/2) * Pi^(n/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 02 2019
EXAMPLE
The a(3) = 60 permutations of the prime indices of A006939(3) = 360: (111223) (121123) (131122) (212113) (231211)
(111232) (121132) (131212) (212131) (232111)
(111322) (121213) (131221) (212311) (311122)
(112123) (121231) (132112) (213112) (311212)
(112132) (121312) (132121) (213121) (311221)
(112213) (121321) (132211) (213211) (312112)
(112231) (122113) (211123) (221113) (312121)
(112312) (122131) (211132) (221131) (312211)
(112321) (122311) (211213) (221311) (321112)
(113122) (123112) (211231) (223111) (321121)
(113212) (123121) (211312) (231112) (321211)
(113221) (123211) (211321) (231121) (322111)
(End)
MAPLE
with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
MATHEMATICA
Table[Apply[Multinomial , Range[n]], {n, 0, 20}] (* Geoffrey Critzer, Dec 09 2012 *) Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *) Table[Length[Permutations[Join@@Table[i, {i, n}, {i}]]], {n, 0, 4}] (* Gus Wiseman, Aug 12 2020 *)
PROG
(PARI) a(n) = binomial(n+1, 2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019
CROSSREFS
A190945 counts the case of anti-run permutations.
A317829 counts partitions of this multiset.
A325617 is the version for factorials instead of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008480 counts permutations of prime indices.
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.
+10
24
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 20, 90, 24, 1, 1, 1, 70, 1680, 2520, 120, 1, 1, 1, 252, 34650, 369600, 113400, 720, 1, 1, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1
COMMENTS
Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2. In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:
m\n:0 1 2 3 4 5
3: 1 6 90 1680 34650 756756 ... A006480; 4: 1 24 2520 369600 63063000 11732745024 ... A008977; 5: 1 120 113400 168168000 305540235000 623360743125120 ... A008978; 6: 1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979; A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).
(End)
EXAMPLE
T(3,5) = (3*5)!/(5!^3) = 756756 = A014609(3) = A006480(5) is the number of permutations of a multiset that contains 3 different elements 5 times, e.g., {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3}.
MATHEMATICA
T[n_, k_]:= (k*n)!/(n!)^k; Table[T[n, k-n], {k, 9}, {n, 0, k-1}]//Flatten
PROG
(Magma) [Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 26 2022 (SageMath)
def A187783(n, k): return gamma(k*(n-k)+1)/(factorial(n-k))^k
Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.
+10
18
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 24, 90, 20, 1, 1, 1, 120, 2520, 1680, 70, 1, 1, 1, 720, 113400, 369600, 34650, 252, 1, 1, 1, 5040, 7484400, 168168000, 63063000, 756756, 924, 1, 1, 1, 40320, 681080400, 137225088000, 305540235000, 11732745024, 17153136, 3432, 1, 1
COMMENTS
T(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1. - Alois P. Heinz, May 06 2013
EXAMPLE
Row n=0: 1, 1, 1, 1, 1, 1, ... A000012Row n=1: 1, 1, 2, 6, 24, 120, ... A000142Row n=2: 1, 1, 6, 90, 2520, 113400, ... A000680Row n=3: 1, 1, 20, 1680, 369600, 168168000, ... A014606Row n=4: 1, 1, 70, 34650, 63063000, 305540235000, ... A014608Row n=5: 1, 1, 252, 756756, 11732745024, 623360743125120, ... A014609
MAPLE
T:= (n, k)-> (k*n)!/(n!)^k:
MATHEMATICA
T[n_, k_] := (k*n)!/(n!)^k; Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 19 2015 *)
a(n) = (n^2)!/(n!)^(n+1); number of ways of dividing n^2 labeled items into n unlabeled boxes of n items each.
+10
11
1, 1, 3, 280, 2627625, 5194672859376, 3708580189773818399040, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000
COMMENTS
Note that if n=p^k for p prime then a(n) is coprime to n (i.e., a(n) is not divisible by p).
a(n) is also the number of labelings for the simple graph K_n X K_n, the graph Cartesian product of the complete graph with itself. - Geoffrey Critzer, Oct 16 2016 a(n) is also the number of knockout tournament seedings with 2 rounds and n participants in each match. - Alexander Karpov, Dec 15 2017
FORMULA
a(n) ~ exp(n - 1/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 23 2018
EXAMPLE
a(2)=3 since the possibilities are {{0,1},{2,3}}; {{0,2},{1,3}}; and {{0,3},{1,2}}.
MAPLE
a:= n-> (n^2)!/(n!)^(n+1):
MATHEMATICA
Table[a[z_] := z^n/n!; (n^2)! Coefficient[Series[a[a[z]], {z, 0, n^2}], z^(n^2)], {n, 1, 10}] (* Geoffrey Critzer, Oct 16 2016 *)
PROG
(PARI) a(n) = (n^2)!/(n!)^(n+1); \\ Altug Alkan, Dec 17 2017
Sum of the terms of the n-th row of triangle pertaining to A096130.
+10
11
1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465
COMMENTS
The product of the terms of the n-th row is given by A034841. Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016
EXAMPLE
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)
MAPLE
seq(add((binomial(n*k, n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007
MATHEMATICA
Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)
PROG
(GAP) List(List([1..20], n->List([1..n], k->Binomial(k*n, n))), Sum); # Muniru A Asiru, Aug 12 2018 (PARI) a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018
CROSSREFS
Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.
Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.
+10
7
1, 1, 2, 1, 6, 6, 1, 20, 90, 24, 1, 70, 1680, 2520, 120, 1, 252, 34650, 369600, 113400, 720, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1, 3432, 17153136, 11732745024, 305540235000, 137225088000, 681080400, 40320, 1, 12870
EXAMPLE
1 1 1 1
2 6 20 70
6 90 1680 34650
24 2520 369600 63063000
PROG
(PARI) T(n, k)=(n*k)!/k!^n;
for(n=1, 6, for(k=1, 6, print1(T(n, k), ", ")); print) \\ Harry J. Smith, Jul 06 2009
G.f.: Sum_{n>=0} (n^2)!/n!^n * x^n / (1-x)^(n^2+1).
+10
7
1, 2, 9, 1714, 63079895, 623361815288736, 2670177752844538217570947, 7363615666255986180456959666126927684, 18165723931631174937747337664794705661513150850379149, 53130688706387570972824498004857476332107293478561950967662962585645710
FORMULA
a(n) = Sum_{k=0..n} Product_{j=0..k-1} binomial(n+j*k,k).
a(n) ~ exp(-1/12) * n^(n^2-n/2+1) / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 23 2013
EXAMPLE
G.f.: A(x) = 1 + 2*x + 9*x^2 + 1714*x^3 + 63079895*x^4 + 623361815288736*x^5 +...
where
A(x) = 1/(1-x) + x/(1-x)^2 + (4!/2!^2)*x^2/(1-x)^5 + (9!/3!^3)*x^3/(1-x)^10 + (16!/4!^4)*x^4/(1-x)^17 + (25!/5!^5)*x^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + x/(1-x)^2 + 6*x^2/(1-x)^5 + 1680*x^3/(1-x)^10 + 63063000*x^4/(1-x)^17 + 623360743125120*x^5/(1-x)^26 +...+ A034841(n)*x^n/(1-x)^(n^2+1) +... Illustrate formula a(n) = Sum_{k=0..n} Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + C(1,1);
a(2) = 1 + C(2,1) + C(2,2)*C(4,2);
a(3) = 1 + C(3,1) + C(3,2)*C(5,2) + C(3,3)*C(6,3)*C(9,3);
a(4) = 1 + C(4,1) + C(4,2)*C(6,2) + C(4,3)*C(7,3)*C(10,3) + C(4,4)*C(8,4)*C(12,4)*C(16,4);
a(5) = 1 + C(5,1) + C(5,2)*C(7,2) + C(5,3)*C(8,3)*C(11,3) + C(5,4)*C(9,4)*C(13,4)*C(17,4) + C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 1 = 2;
a(2) = 1 + 2 + 1*6 = 9;
a(3) = 1 + 3 + 3*10 + 1*20*84 = 1714;
a(4) = 1 + 4 + 6*15 + 4*35*120 + 1*70*495*1820 = 63079895;
a(5) = 1 + 5 + 10*21 + 10*56*165 + 5*126*715*2380 + 1*252*3003*15504*53130 = 623361815288736; ...
MAPLE
with(combinat):
a:= n-> add(multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
MATHEMATICA
Table[Sum[Product[Binomial[n+j*k, k], {j, 0, k-1}], {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 23 2013 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*x^m/(1-x+x*O(x^n))^(m^2+1)), n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, prod(j=0, k-1, binomial(n+j*k, k)))}
for(n=0, 15, print1(a(n), ", "))
a(n) is the least integer of the form (n^2)!/(n!)^k.
+10
5
1, 3, 280, 2627625, 5194672859376, 5150805819130303332, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000
COMMENTS
(n^2)!/(n!)^(n+1) is an integer for every n (see A057599). Hence k >= n+1. Conjecture: k=n+1 only when n is prime or a power of a prime.
EXAMPLE
a(4) = 16!/(4!)^5 = 2627625 which is not further divisible by 24.
PROG
(PARI) a(n)={if(n==1, 1, (n^2)!/(n!^valuation((n^2)!, n!)))} \\ Andrew Howroyd, Nov 09 2019
Number of sequences with up to n copies each of 1,2,...,n.
+10
5
1, 2, 19, 5248, 191448941, 1856296498826906, 7843008902239185171370147, 21408941228439913825832318523364743824, 52400635808473472283994952631626957015306076632624953, 152306240915343870544748050434914720360496623911547121447677238156864610
FORMULA
a(n) ~ exp(11/12) * n^(n^2 - n/2 + 1) / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, May 24 2020
EXAMPLE
a(0) = 1: () = the empty sequence.
a(1) = 2: (), 1.
a(2) = 19: (), 1, 2, 11, 12, 21, 22, 112, 121, 122, 211, 212, 221, 1122, 1212, 1221, 2112, 2121, 2211.
MAPLE
b:= proc(n, k, i) option remember; `if`(k=0, 1,
`if`(i<1, 0, add(b(n, k-j, i-1)/j!, j=0..min(k, n))))
end:
a:= n-> add(b(n, k, n)*k!, k=0..n^2):
seq(a(n), n=0..10);
MATHEMATICA
Table[Sum[k!*SeriesCoefficient[Sum[x^j/j!, {j, 0, n}]^n, {x, 0, k}], {k, 0, n^2}], {n, 0, 10}] (* Vaclav Kotesovec, May 24 2020 *)
PROG
(PARI) {a(n) = sum(i=0, n^2, i!*polcoef(sum(j=0, n, x^j/j!)^n, i))} \\ Seiichi Manyama, May 19 2019
Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).
+10
3
0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 4, 4, 2, 0, 0, 3, 6, 10, 6, 3, 0, 0, 3, 9, 16, 16, 9, 3, 0, 0, 4, 12, 28, 32, 28, 12, 4, 0, 0, 4, 16, 40, 60, 60, 40, 16, 4, 0, 0, 5, 20, 60, 100, 126, 100, 60, 20, 5, 0, 0, 5, 25, 80, 160, 226, 226, 160, 80, 25, 5, 0, 0, 6, 30, 110, 240
COMMENTS
Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.
Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - Yosu Yurramendi, Aug 08 2008 T(n, k) is the sum of odd-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for A034851(n, k) paths and odd for T(n, k) of them. For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for A034851(n, k) strings and odd for T(n, k) cases. (End)
FORMULA
T(n, k) = T(n - 1, k - 1) + T(n - 1, k); except when n is even and k odd, in which case T(n, k) = A034851(n, k) = T(n - 1, k - 1) + A034841(n - 1, k) = A034841(n - 1, k - 1) + T(n - 1, k) = C(n, k) / 2. - Álvar Ibeas, Jun 01 2020
EXAMPLE
Triangle begins:
0;
0 0;
0 1 0;
0 1 1 0;
0 2 2 2 0;
0 2 4 4 2 0;
...
MATHEMATICA
nmax = 12; t[n_?EvenQ, k_?EvenQ] := (Binomial[n, k] - Binomial[n/2, k/2])/ 2; t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_?OddQ, k_?EvenQ] := (Binomial[n, k] - Binomial[(n-1)/2, k/2])/2; t[n_?OddQ, k_?OddQ] := (Binomial[n, k] - Binomial[(n-1)/2, (k-1)/2])/2; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 15 2011, after Yosu Yurramendi *)
PROG
(Haskell)
a034852 n k = a034852_tabl !! n !! k
a034852_row n = a034852_tabl !! n
a034852_tabl = zipWith (zipWith (-)) a007318_tabl a034851_tabl
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