A395216 a(n) = (2*n)! * [x^(2*n)] 1/sqrt((1 - x)*(1 - 3*x)).

1, 9, 681, 148905, 64805265, 46896669225, 50831084252025, 77061447551313225, 155672917524626126625, 404153655359180645543625, 1311161072449806104077175625, 5197766162779913140792304555625, 24721993179513170562351592437800625, 138944861938899421852917299018079515625

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

Let A(n,k) = (2*n)! * [x^(2*n)] 1/((1 - x)*(1 - 3*x))^(k/2). A(0,k) = 1 and A(n,k) = k*(k+2) * A(n-1,k+4) + 3*k*(k+1) * A(n-1,k+2) for n > 0. a(n) = A(n,1).

a(n) = (2*n)! * Sum_{k=0..n} 16^k * (-3)^(n-k) * binomial(n+k-1/2,n+k) * binomial(n+k,2*k).

a(n) ~ 2^(2*n) * 3^(2*n + 1/2) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Apr 30 2026

Mathematica

Table[(-3)^n * Binomial[n - 1/2, n] * (2*n)! * Hypergeometric2F1[-n, 1/2 + n, 1/2, 4/3], {n, 0, 20}] (* Vaclav Kotesovec, Apr 30 2026 *)

Prog

(PARI) a(n) = my(x='x+O('x^(2*n+1))); (2*n)!*polcoef(1/sqrt((1-x)*(1-3*x)), 2*n);

Crossrefs

Column k=1 of A395215.

Sequence in context: A372184 A053973 A059492 * A322488 A109061 A332169

Adjacent sequences: A395213 A395214 A395215 * A395217 A395218 A395219

Keyword

nonn

Author

Seiichi Manyama, Apr 16 2026

Status

approved