A360144 a(n) = Sum_{k=0..n} binomial(2*n+3*k,n-k).

1, 3, 14, 69, 344, 1721, 8621, 43206, 216570, 1085574, 5441294, 27272044, 136679882, 684959516, 3432431414, 17199626276, 86182614207, 431824008713, 2163629549132, 10840520569183, 54313805146415, 272122594209738, 1363372115057995, 6830627007245263

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^5) ), where c(x) is the g.f. of A000108.

D-finite with recurrence +n*(697*n-7543)*a(n) +(697*n^2+23641*n-3800)*a(n-1) +2*(-32006*n^2+199879*n-255053)*a(n-2) +(283953*n^2-2288641*n+4072186)*a(n-3) +2*(-186566*n^2+1774989*n-4013515)*a(n-4) +(146221*n^2-1648033*n+4472550)*a(n-5) +(38223*n^2-307771*n+532906)*a(n-6) -10*(1511*n-6875)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Mar 12 2023

a(n) = binomial(2*n, n)*hypergeom([1, (1+2*n)/3, 2*(1+n)/3, 1+2*n/3, -n], [(1+n)/4, (2+n)/4, (3+n)/4, 1+n/4], -3^3/4^4). - Stefano Spezia, Jun 17 2025

Recurrence (of order 5): (n-4)*n*(7*n^3 - 98*n^2 + 435*n - 624)*a(n) = 2*(35*n^5 - 637*n^4 + 4345*n^3 - 13631*n^2 + 18708*n - 7560)*a(n-1) - (203*n^5 - 3738*n^4 + 26503*n^3 - 90288*n^2 + 147120*n - 90720)*a(n-2) + 10*(n-3)*(2*n - 9)*(7*n^3 - 77*n^2 + 260*n - 280)*a(n-3) + (n-4)*n*(7*n^3 - 98*n^2 + 435*n - 624)*a(n-4) - 2*(n-3)*(2*n - 9)*(7*n^3 - 77*n^2 + 260*n - 280)*a(n-5). - Vaclav Kotesovec, Nov 09 2025

Maple

A360144 := proc(n)
    add(binomial(2*n+3*k, n-k), k=0..n) ;
end proc:
seq(A360144(n), n=0..70) ; # R. J. Mathar, Mar 12 2023

Prog

(PARI) a(n) = sum(k=0, n, binomial(2*n+3*k, n-k));
(PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^5)))

Crossrefs

Cf. A001700, A032443, A108080, A360143.

Cf. A000108, A000344, A000984.

Sequence in context: A121185 A276904 A199314 * A190725 A151325 A020065

Adjacent sequences: A360141 A360142 A360143 * A360145 A360146 A360147

Keyword

nonn

Author

Seiichi Manyama, Jan 27 2023

Status

approved