A369688 G.f. satisfies A(x) = 1 + x*A(x) + x^2*(1-x)^3*A(x)^5.

1, 1, 2, 4, 12, 36, 126, 442, 1644, 6172, 23801, 92731, 366688, 1462852, 5891808, 23898576, 97600556, 400844140, 1654818768, 6862550360, 28576414261, 119434041561, 500849380048, 2106740001442, 8886482895068, 37580609774876, 159303913630686

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1532

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * binomial(5*k,k) / (4*k+1).

D-finite with recurrence: 2869*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n) - 2*(5482*n + 19443)*(n + 4)*(n + 3)*(n + 2)*a(n + 1) + (n + 4)*(n + 3)*(14910*n^2 + 121840*n + 249203)*a(n + 2) - (n + 4)*(7380*n^3 + 104750*n^2 + 490432*n + 759411)*a(n + 3) - (n + 4)*(715*n^3 + 6320*n^2 + 11412*n - 11904)*a(n + 4) + 256*(2*n + 11)*(n + 5)*(3*n^2 + 31*n + 81)*a(n + 5) - 64*(n + 5)*(2*n + 13)*(2*n + 11)*(n + 6)*a(n + 6) = 0. - Robert Israel, May 01 2026

Maple

f:= gfun:-rectoproc({2869*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n) - 2*(5482*n + 19443)*(n + 4)*(n + 3)*(n + 2)*a(n + 1) + (n + 4)*(n + 3)*(14910*n^2 + 121840*n + 249203)*a(n + 2) - (n + 4)*(7380*n^3 + 104750*n^2 + 490432*n + 759411)*a(n + 3) - (n + 4)*(715*n^3 + 6320*n^2 + 11412*n - 11904)*a(n + 4) + 256*(2*n + 11)*(n + 5)*(3*n^2 + 31*n + 81)*a(n + 5) - 64*(n + 5)*(2*n + 13)*(2*n + 11)*(n + 6)*a(n + 6), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 12, a(5) = 36}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 01 2026

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n, 2*k)*binomial(5*k, k)/(4*k+1));

Crossrefs

Cf. A227035, A346647, A364522, A364597.

Cf. A049130, A364594.

Sequence in context: A148209 A148210 A148211 * A009152 A081443 A086647

Adjacent sequences: A369685 A369686 A369687 * A369689 A369690 A369691

Keyword

nonn

Author

Seiichi Manyama, Jan 28 2024

Status

approved