A291898 (n+1)^2*a(n+1) = -(9*n^2 + 9*n + 3)*a(n) - 27*n^2*a(n-1), with a(0) = 1 and a(1) = -3.

1, -3, 9, -21, 9, 297, -2421, 12933, -52407, 145293, -35091, -2954097, 25228971, -142080669, 602217261, -1724917221, 283305033, 38852066421, -337425235479, 1938308236731, -8364863310291, 24286959061533, -3011589296289, -574023003011199, 5028616107443691

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1400

Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.

Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.

Zhi-Hong Sun, Super congruences concerning binomial coefficients and Apéry-like numbers, arXiv:2002.12072 [math.NT], 2020.

Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.

Formula

Given A(x) is the g.f. of this sequence, B(x) is the g.f. of A005928, and C(x) is the g.f. of A121589, then B(x) = A(C(x)).

a(n) = (-1)^n * A006077(n).

0 = y*(3 + 27*x) + y'*(1 + 18*x + 81*x^2) + y''*(x + 9*x^2 + 27*x^3) where y(x) is the g.f. of this sequence.

Example

G.f. = 1 - 3*x + 9*x^2 - 21*x^3 + 9*x^4 + 297*x^5 - 2421*x^6 + ...

Maple

f:= gfun:-rectoproc({(n+1)^2*a(n+1) = -(9*n^2 + 9*n + 3)*a(n) - 27*n^2*a(n-1), a(0)=1, a(1)=-3}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Nov 02 2017

Mathematica

a[ n_] := If[ n < 0, 0, (-3)^n HypergeometricPFQ[ {-n, 1 - n, 2 - n}/3, {1, 1}, 1]];
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1, 2}/3, {1}, (3 x / (1 + 3 x))^3 ] / (1 + 3 x), {x, 0, n}];
nxt[{n_, a_, b_}]:={n+1, b, (-(9 n^2+9n+3)b- a 27n^2)/(n+1)^2}; NestList[nxt, {1, 1, -3}, 30][[;; , 2]] (* Harvey P. Dale, Nov 02 2024 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( subst(eta(x + A)^3 / eta(x^3 + A), x, serreverse( x * eta(x^9 + A)^3 / eta(x + A)^3)), n))};
(Magma) I:=[-3, 9]; [1] cat [n le 2 select I[n] else (-1)*((9*n^2-9*n+3)*Self(n-1) + 27*(n-1)^2*Self(n-2))/n^2: n in [1..30]]; // G. C. Greubel, Jul 28 2018

Crossrefs

Cf. A005928, A006077, A121589.

Sequence in context: A305618 A377947 A006077 * A330987 A365919 A109612

Adjacent sequences: A291895 A291896 A291897 * A291899 A291900 A291901

Keyword

sign

Author

Michael Somos, Nov 02 2017

Status

approved