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
Keywords
Programs
-
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
-
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)))
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