A344500 a(n) = Sum_{k=0..n} binomial(n, k)*CT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*CT(n, k), where CT(n, k) is the Catalan triangle A053121.

1, 1, 2, 7, 21, 66, 216, 715, 2395, 8101, 27598, 94568, 325612, 1125632, 3904512, 13583195, 47373255, 165585883, 579907758, 2034443127, 7148313381, 25151582046, 88607951512, 312518438532, 1103393962996, 3899415207676, 13792683831176, 48825746365672, 172971084083752

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Formula

a(n) = Sum_{j=0..n} even(n + j)*binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1), where even(k) = 1 if k is even and otherwise 0.

From Vaclav Kotesovec, Apr 21 2024: (Start)

Recurrence: 2*(n+1)*(2*n + 1)*(13*n^3 - 17*n^2 - 12*n + 10)*a(n) = (143*n^5 - 44*n^4 - 525*n^3 + 320*n^2 + 94*n - 60)*a(n-1) + 4*(n-1)*(26*n^4 + 5*n^3 - 108*n^2 + 7*n + 22)*a(n-2) + 16*(n-2)*(n-1)*(13*n^3 + 22*n^2 - 7*n - 6)*a(n-3).

a(n) ~ sqrt((247 - 9131*13^(2/3)/(1788163 + 409728*sqrt(78))^(1/3) + (13*(1788163 + 409728*sqrt(78)))^(1/3))/13) * ((11 + (1/3)*(172017 - 16848*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^n / (sqrt(Pi*n) * 2^(2*n + 3) * 3^(n + 1/2))). (End)

Maple

a := n -> add((if n + j mod 2 = 1 then 0 else binomial(n, j)*binomial(n + 1, (n - j)/2)*(j + 1)/(n + 1) fi), j = 0..n): seq(a(n), n = 0..28);

Mathematica

Table[Sum[(1 + (-1)^(n+j))/2 * Binomial[n, j] * Binomial[n+1, (n-j)/2] * (j+1)/(n+1), {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 21 2024 *)

Crossrefs

Cf. A053121, A344501.

Sequence in context: A331722 A052911 A126133 * A186240 A274203 A330058

Adjacent sequences: A344497 A344498 A344499 * A344501 A344502 A344503

Keyword

nonn,changed

Author

Peter Luschny, May 22 2021

Status

approved