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a(n) = 6*(n+1)*(2*n+6)!/((n+3)!*(n+5)!).
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0, 6, 28, 108, 396, 1430, 5148, 18564, 67184, 244188, 891480, 3268760, 12034980, 44482230, 165002460, 614106900, 2292665760, 8583849780, 32223863880, 121267584360, 457412818200, 1729020452796, 6548744132568, 24849948274088, 94460672942496, 359656297841400
COMMENTS
a(n) is the number of permutations pi of [n+4] such that s(pi) is one of 132456...(n+4), 231456...(n+4), 312456...(n+4), or 321456...(n+4) and s denotes West's stack-sorting map. - Colin Defant, Jan 14 2019
FORMULA
G.f.: (2 - 9*x + 6*x^2 + 2*x^3 + (5*x-2)*sqrt(1-4*x))/(x^4) = 6*x + 14*x^2*G(0); G(k) = 1 + 1/(k + 1 - 2*x*(k+1)*(k+2)*(2*k+9)/(2*x*(k+2)*(2*k+9) + (k+7)/G(k+1))); (continued fraction, 3-step ). - Sergei N. Gladkovskii, Jan 08 2012 -(n+4)*(n-1)*a(n) + 2*n*(2*n+3)*a(n-1) = 0, n > 0. - R. J. Mathar, Dec 15 2015 G.f.: x*C(x)^6 + 3*(1+C(x))C(x)^4, where C(x) = (1-sqrt(1-4x))/(2x) is the Catalan function. - Colin Defant, Jan 14 2019
MATHEMATICA
Table[6 (n + 1) (2 n + 6)! / ((n + 3)! (n + 5)!), {n, -1, 25}] (* Vincenzo Librandi, Jan 15 2019 *)
PROG
(Magma) [6*(n+1)*Factorial(2*n+6)/(Factorial(n+3)*Factorial(n+5)): n in [-1..40]]; // Vincenzo Librandi, Jan 15 2019
Triangle read by rows: T(n,k) is the number of binary trees with n edges and such that the first leaf in the preorder traversal is at level k (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
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2, 1, 4, 2, 4, 8, 5, 9, 12, 16, 14, 24, 30, 32, 32, 42, 70, 85, 88, 80, 64, 132, 216, 258, 264, 240, 192, 128, 429, 693, 819, 833, 760, 624, 448, 256, 1430, 2288, 2684, 2720, 2490, 2080, 1568, 1024, 512, 4862, 7722, 9009, 9108, 8361, 7068, 5488, 3840, 2304, 1024
COMMENTS
Row sums are the Catalan numbers ( A000108). T(n,1)= A000108(n-1) for n>=2 (the Catalan numbers). T(n,n)=2^n. Sum(k*T(n,k),k=1..n)= A120989(n).
FORMULA
T(n,k)=Sum(j*binomial(k,j)*binomial(2n-2k+j,n-k)/(2n-2k+j), j=0..k). G.f.=1/[1-tz(1+C)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
EXAMPLE
T(2,1)=1 because we have the tree /\.
Triangle starts:
2;
1;4;
2,4,8;
5,9,12,16;
14,24,30,32,32;
MAPLE
T:=proc(n, k) if k<n then add(j*binomial(k, j)*binomial(2*n-2*k+j, n-k)/(2*n-2*k+j), j=0..k) elif k=n then 2^n else 0 fi end:for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
Number of North-East lattice paths from (0,0) to (n,n) that have exactly three east steps below the subdiagonal y = x-1.
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5, 24, 95, 356, 1309, 4784, 17472, 63920, 234498, 863056, 3187041, 11807740, 43885725, 163601760, 611625660, 2292665760, 8615485590, 32451382800, 122499978510, 463369822344, 1756113365874, 6667436894624, 25357090075600, 96589604043296, 368478056090340, 1407687015207200, 5384924914890213
COMMENTS
This sequence is related to paired pattern P_1 in Pan and Remmel's link.
FORMULA
G.f.: -((-1 + f(x) + 2*x)^2*(-1 + f(x) + 2*(f(x) - 2*x)*x))/(8*x^2), where f(x) = sqrt(1 - 4*x).
Triangle read by rows related to Catalan triangle A009766.
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1, 1, 1, 2, 3, 2, 5, 9, 9, 5, 14, 28, 34, 28, 14, 42, 90, 123, 123, 90, 42, 132, 297, 440, 497, 440, 297, 132, 429, 1001, 1573, 1935, 1935, 1573, 1001, 429, 1430, 3432, 5642, 7397, 8068, 7397, 5642, 3432, 1430, 4862, 11934, 20332, 28014, 32636, 32636, 28014, 20332, 11934
LINKS
Laurent Méhats, Lutz Straßburger, Non-crossing Tree Realizations of Ordered Degree Sequences, Pages 211-227 in Logical Aspects of Computational Linguistics. Celebrating 20 Years of LACL (1996-2016), 9th International Conference, LACL 2016, Nancy, France, December 5-7, 2016, Proceedings, Lecture Notes in Computer Science book series (LNCS, volume 10054). See Eq. (7).
EXAMPLE
Triangle begins:
1,
1,1,
2,3,2,
5,9,9,5,
14,28,34,28,14,
42,90,123,123,90,42,
132,297,440,497,440,297,132,
...
MAPLE
if q <0 then
0;
else
binomial(2*q, q)/(1+q) ;
end if;
end proc:
R := proc(q, s)
option remember;
local a, j, l ;
if q= 0 then
elif s = 0 then
else
a := 0 ;
for j from 0 to q do
for l from 0 to s do
if j+l-1 >= 0 then
a := a+ A000108(j+l-1) *procname(q-j, s-l) ; end if;
end do:
end do:
end if;
end proc:
R(n-k, k) ;
end proc:
MATHEMATICA
R[q_, s_] := R[q, s] = Module[{a, j, l}, If[q == 0, CatalanNumber[s], If[s == 0, CatalanNumber[q], a = 0; For[j = 0, j <= q, j++, For[l = 0, l <= s , l++, If[j + l - 1 >= 0, a = a + CatalanNumber[j + l - 1] R[q - j, s - l]] ]]]] /. Null -> a];
T [n_, k_] := R[n - k, k];
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