A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).

1, 0, -1, 3, -6, 6, 15, -107, 349, -672, 39, 5835, -27654, 75765, -95799, -279129, 2297970, -8377854, 17663640, -996624, -177445221, 888491025, -2551959604, 3337931168, 10407149226, -87719805853, 328682535695, -708428979213, 15252552804, 7616368090377, -38693979668535

Offset

0,4

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).

D-finite with recurrence 2*n*(2*n+1)*a(n) +(51*n-26)*(n-1)*a(n-1) +(279*n^2 -931*n +766)*a(n-2) +2*(413*n^2 -2127*n +2728)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023

From Peter Bala, Aug 25 2024: (Start)

Fifth-order recurrence: 2*(n-2)*(n-1)*n*(2*n+1)*a(n) + (n-2)*(n-1)*(31*n^2-27*n+6)*a(n-1) + 3*(n-2)*(3*n-5)*(12*n^2-19*n+2)*a(n-2) + 3*(3*n-8)*(18*n^3-69*n^2+63*n-2)*a(n-3) + 3*n*(3*n-11)*(12*n^2-49*n+42)*a(n-4) + 3*n*(n-1)*(3*n-10)*(3*n-14)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 3 and a(4) = -6.

The g.f. A(x) satisfies (1/x) * series_reversion(x/A(x)) = 1 - x^2 + 3*x^3 - 4*x^4 - 9*x^5 + 73*x^6 - ..., the g.f. of A364376. (End)

Maple

A364372 := proc(n)
    add( (-1)^k*binomial(3*k+1, k) * binomial(3*k+1, n-k)/(3*k+1), k=0..n) ;
end proc:
seq(A364372(n), n=0..80); # R. J. Mathar, Jul 25 2023

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*k+1, k)*binomial(3*k+1, n-k)/(3*k+1));

Crossrefs

Cf. A364371, A364373, A364376.

Cf. A364336.

Sequence in context: A238775 A269525 A341885 * A036252 A103463 A370154

Adjacent sequences: A364369 A364370 A364371 * A364373 A364374 A364375

Keyword

sign,easy

Author

Seiichi Manyama, Jul 20 2023

Status

approved