A241023 Central terms of the triangle in A102413.

1, 4, 16, 76, 384, 2004, 10672, 57628, 314368, 1728292, 9560016, 53144172, 296642688, 1661529588, 9333781872, 52566230076, 296697618432, 1677889961028, 9505151782288, 53928746972812, 306393243712384, 1742920028025364, 9925790375394096, 56584659163097436

Offset

0,2

Comments

a(n) = A102413(2*n,n).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Formula

a(n) = 4 * A047781(n).

a(n) = 2*Hyper2F1([-n, n], [1], -1) for n>0. - Peter Luschny, Aug 02 2014

D-finite g.f. = (1+x)/sqrt(1-6*x+x^2), pairwise sums of A001850. - R. J. Mathar, Jan 15 2020

From Peter Bala, Apr 16 2024: (Start)

a(n) = Sum_{k = 0..n} (-1)^(n-k)*(2^k)*binomial(2*k, k)*binomial(n+k-1, n-k).

a(n) = (-1)^(n+1) * 4*n * hypergeom([n+1, -n+1], [2], 2).

n*(2*n - 3)*a(n) = 4*(3*n^2 - 6*n + 2)*a(n-1) - (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1 and a(1) = 4.

O.g.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)*x^n/(1 + x)^(2*n) = 1 + 4*x + 16*x^2 + 76*x^3 + 384*x^4 + .... (End)

Mathematica

a[0] = 1; a[n_] := 4 Hypergeometric2F1[1 - n, n + 1, 1, -1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 28 2019 *)

Prog

(Haskell)
a241023 n = a102413 (2 * n) n

Crossrefs

Cf. A001850, A047781, A102413.

Sequence in context: A094559 A199214 A374566 * A200725 A255906 A260949

Adjacent sequences: A241020 A241021 A241022 * A241024 A241025 A241026

Keyword

nonn,easy

Author

Reinhard Zumkeller, Apr 15 2014

Status

approved