A318109 a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k.

1, 4, 46, 652, 10186, 168304, 2884456, 50723824, 909192538, 16538659384, 304391739796, 5655971294824, 105929883322576, 1997228410630912, 37871584674309376, 721672204654077952, 13811327854028171098, 265324110145941691912, 5114208160758838538044, 98874597697991698311832, 1916741738060370782929036

Offset

0,2

Comments

Diagonal of rational function 1/(1 - (x + y + z - 2*x*y*z)).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..766 (terms 0..100 from Gheorghe Coserea)

Formula

G.f. y=A(x) satisfies: 0 = x*(x - 1)*(4*x - 1)*(8*x^2 + 20*x - 1)*y'' + (96*x^4 + 64*x^3 - 120*x^2 + 42*x - 1)*y' + 4*(2*x + 1)*(4*x^2 - 2*x + 1)*y.

From Peter Bala, Mar 16 2023: (Start)

n^2*(3*n - 4)*a(n) = (3*n - 2)*(21*n^2 - 35*n + 10)*a(n-1) - 4*(9*n^3 - 30*n^2 + 29*n - 6)*a(n-2) - 8*(3*n - 1)*(n - 2)^2*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 46.

Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. (End)

a(n) ~ (1 + sqrt(3))^(3*n + 1) / (2*Pi*sqrt(3)*n). - Vaclav Kotesovec, Mar 17 2023

Example

A(x) = 1 + 4*x + 46*x^2 + 652*x^3 + 10186*x^4 + 168304*x^5 + 2884456*x^6 + ...

Prog

(PARI)
a(n) = sum(k=0, n, (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k);
vector(21, n, a(n-1))

Crossrefs

Cf. A000172, A124435, A318107, A318108.

Sequence in context: A236956 A113264 A264717 * A234527 A126739 A191870

Adjacent sequences: A318106 A318107 A318108 * A318110 A318111 A318112

Keyword

nonn,easy

Author

Gheorghe Coserea, Sep 20 2018

Status

approved