A365728 G.f. satisfies A(x) = 1 + x^4*A(x)^2*(1 + x*A(x)).

1, 0, 0, 0, 1, 1, 0, 0, 2, 5, 3, 0, 5, 21, 28, 12, 14, 84, 180, 165, 97, 330, 990, 1430, 1133, 1560, 5005, 10010, 11349, 11193, 25452, 61880, 94250, 100844, 150144, 360468, 683162, 889542, 1100784, 2144397, 4536839, 7158326, 9102786, 14132580, 29078645

Offset

0,9

Links

Robert Israel, Table of n, a(n) for n = 0..4364

Formula

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

D-finite with recurrence: (-16*n^2 - 48*n - 32)*a(n) + (-108*n^2 - 648*n - 924)*a(n + 2) + (-12*n^2 + 180*n + 1056)*a(n + 3) + (16*n^2 + 216*n + 680)*a(n + 4) + (405*n^2 + 6075*n + 22500)*a(n + 5) + (351*n^2 + 5913*n + 24732)*a(n + 6) + (55*n^2 + 999*n + 4508)*a(n + 7) + (-3*n^2 - 63*n - 324)*a(n + 8) + (-60*n^2 - 1350*n - 7500)*a(n + 10) + (-12*n^2 - 306*n - 1944)*a(n + 11) = 0. - Robert Israel, Mar 02 2026

Maple

f:= gfun:-rectoproc({(-16*n^2 - 48*n - 32)*a(n) + (-108*n^2 - 648*n - 924)*a(n + 2) + (-12*n^2 + 180*n + 1056)*a(n + 3) + (16*n^2 + 216*n + 680)*a(n + 4) + (405*n^2 + 6075*n + 22500)*a(n + 5) + (351*n^2 + 5913*n + 24732)*a(n + 6) + (55*n^2 + 999*n + 4508)*a(n + 7) + (-3*n^2 - 63*n - 324)*a(n + 8) + (-60*n^2 - 1350*n - 7500)*a(n + 10) + (-12*n^2 - 306*n - 1944)*a(n + 11), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1, a(5) = 1, a(6) = 0, a(7) = 0, a(8) = 2, a(9) = 5, a(10) = 3}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 02 2026

Prog

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

Crossrefs

Cf. A365727, A365729, A365730, A365731.

Cf. A365696.

Sequence in context: A011035 A102892 A132898 * A265318 A279536 A269954

Adjacent sequences: A365725 A365726 A365727 * A365729 A365730 A365731

Keyword

nonn

Author

Seiichi Manyama, Sep 17 2023

Status

approved