A322244 G.f.: 1/sqrt(1 - 6*x - 55*x^2).

1, 3, 41, 315, 3345, 31923, 328889, 3337323, 34600225, 359225955, 3760299081, 39497556123, 416692693041, 4409256847635, 46791791441625, 497734241873355, 5305782027097665, 56663444325365955, 606142658305541225, 6493612892317230075, 69658589316520324945, 748141936546712050035, 8043908203413946807545, 86573015247061060850475, 932597459464760512144225

Offset: 0

Author

Paul D. Hanna, Dec 09 2018

Keywords

• nonn
• easy

Examples

			G.f.: A(x) = 1 + 3*x + 41*x^2 + 315*x^3 + 3345*x^4 + 31923*x^5 + 328889*x^6 + 3337323*x^7 + 34600225*x^8 + 359225955*x^9 + 3760299081*x^10 + ...
such that A(x)^2 = 1/(1 - 6*x - 55*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 3*x + 25*x^2 + 171*x^3 + 1457*x^4 + 12243*x^5 + 109769*x^6 + 997755*x^7 + 9314657*x^8 + 88177059*x^9 + 847159161*x^10 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..962

Crossrefs

Cf. A322245 (a(n)^2).

Programs

Mathematica

a[n_] := Sum[(-5)^(n-k) * 4^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)
CoefficientList[Series[1/Sqrt[1-6x-55x^2],{x,0,40}],x] (* Harvey P. Dale, Aug 13 2024 *)

PARI

/* Using generating function: */
{a(n) = polcoeff( 1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

PARI

/* Using binomial formula: */
{a(n) = sum(k=0,n, (-5)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

PARI

/* Using binomial formula: */
{a(n) = sum(k=0,n, 11^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

PARI

/* a(n) is central coefficient in (1 + 3*x + 4*x^2)^n */
{a(n) = polcoeff( (1 + 3*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).

a(n) = Sum_{k=0..n} (-5)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).

a(n) equals the (central) coefficient of x^n in (1 + 3*x + 16*x^2)^n.

a(n) ~ 11^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 13 2018

D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +55*(-n+1)*a(n-2)=0. - R. J. Mathar, Jan 16 2020

a(n) = (1/4)^n * Sum_{k=0..n} (-5)^k * 11^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025