| Displaying 1-4 of 4 results found.
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1
1, 11, 73, 403, 2021, 9567, 43611, 193683, 844213, 3629083, 15437951, 65143503, 273148279, 1139548469, 4734740493, 19606960755, 80969809797, 333601494651
FORMULA
a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i ( A056040).
MAPLE
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i, k; add(add((-1)^(n-i)*binomial(n-k, n-i)*swing(2*i+1), i=k..n), k=0..n) end:
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)* Binomial[n - k, n - i]*sf[2*i + 1], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)
Triangle interpolating the swinging factorial ( A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients ( A002426) with the central binomial coefficients ( A000984).
+10
6
1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870
COMMENTS
Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i ( A056040). This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019
EXAMPLE
Triangle begins
1;
1, 2;
3, 4, 6;
7, 10, 14, 20;
19, 26, 36, 50, 70;
51, 70, 96, 132, 182, 252;
141, 192, 262, 358, 490, 672, 924;
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts: 1 1 3 7 19 51 ...
2 4 10 26 70 192 ...
6 14 36 96 262 726 ...
20 50 132 358 988 2760 ...
70 182 490 1346 3748 10540 ...
252 672 1836 5094 14288 40404 ...
(...)
Read by falling antidiagonals this yields the same sequence. (End)
MAPLE
For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle. a := n -> DiffTria(k->swing(2*k), n, true);
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Triangle read by rows interpolating the swinging subfactorial ( A163650) with the swinging factorial ( A056040).
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4
1, 0, 1, 1, 1, 2, 2, 3, 4, 6, -9, -7, -4, 0, 6, 44, 35, 28, 24, 24, 30, -165, -121, -86, -58, -34, -10, 20, 594, 429, 308, 222, 164, 130, 120, 140, -2037, -1443, -1014, -706, -484, -320, -190, -70, 70, 6824, 4787, 3344, 2330, 1624, 1140, 820, 630, 560, 630
COMMENTS
An analog to the derangement triangle ( A068106).
FORMULA
T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i ( A056040).
EXAMPLE
1
0, 1
1, 1, 2
2, 3, 4, 6
-9, -7, -4, 0, 6
44, 35, 28, 24, 24, 30
-165, -121, -86, -58, -34, -10, 20
MAPLE
DiffTria := proc(f, n, display) local m, A, j, i, T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] - A[j] od;
for i from 0 to m do T := T, (-1)^(m-i)*A[i] od;
if display then print(seq(T[i], i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL, [T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
Computes n rows of the triangle.
A163770 := n -> DiffTria(k->swing(k), n, true);
A068106 := n -> DiffTria(k->factorial(k), n, true);
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) ( A002457).
+10
2
1, 5, 19, 67, 227, 751, 2445, 7869, 25107, 79567, 250793, 786985, 2460397, 7667921, 23832931, 73902627, 228692115, 706407903, 2178511449, 6708684009, 20632428249, 63380014845, 194486530791, 596213956023, 1826103432573, 5588435470401, 17089296473655
COMMENTS
Also a(n) = sum {i=0..n} (-1)^(n-i) binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i ( A056040).
FORMULA
O.g.f.: A(x)=1/(1-x*M(x))^3, M(x) - o.g.f. of A001006. a(n) = sum(k^3/n *sum(C(n,j)*C(j,2*j-n-k), j=0..n), k=1..n). - Vladimir Kruchinin, Sep 06 2010 Recurrence: n*a(n) = (2*n+3)*a(n-1) + 3*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 21 2012 a(n) = (-1)^n*hypergeom([-n,3/2], [1], 4). - Peter Luschny, Apr 26 2016
MAPLE
a := proc(n) local i; add((-1)^(n-i)*binomial(n, i)/Beta(i+1, i+1), i=0..n) end:
seq(simplify((-1)^n*hypergeom([-n, 3/2], [1], 4)), n=0..26); # Peter Luschny, Apr 26 2016
MATHEMATICA
CoefficientList[Series[Sqrt[x+1]/(1-3*x)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 21 2012 *) sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[(-1)^(n-i)*Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)
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