A322248 G.f.: 1/sqrt( (1 + 3*x)*(1 - 13*x) ).

1, 5, 57, 605, 6961, 81525, 973545, 11765325, 143522145, 1763351525, 21789466777, 270509191485, 3371353189009, 42155188480085, 528587607974217, 6644129071092525, 83691484792766145, 1056178325362832325, 13351036742005533945, 169019946403985898525, 2142600388730167543281, 27193744661180635582005, 345520219114720175821737, 4394534009569783690837005, 55943630366450131877449761, 712778930909503993783945125

Offset: 0

Author

Paul D. Hanna, Dec 10 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 5*x + 57*x^2 + 605*x^3 + 6961*x^4 + 81525*x^5 + 973545*x^6 + 11765325*x^7 + 143522145*x^8 + 1763351525*x^9 + 21789466777*x^10 + ...
such that A(x)^2 = 1/(1 - 10*x - 39*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 41*x^2 + 365*x^3 + 3537*x^4 + 35925*x^5 + 378105*x^6 + 4084925*x^7 + 45044129*x^8 + 504880805*x^9 + 5735247817*x^10 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..899
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Crossrefs

Cf. A084605, A098409, A248168.

Cf. A322249.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[((1+3x)(1-13x))],{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)

PARI

/* Using generating function: */
{a(n) = polcoeff( 1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

PARI

/* Using binomial formula: */
{a(n) = sum(k=0,n, (-3)^(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, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

PARI

/* a(n) as a central coefficient */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).

a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).

a(n) equals the (central) coefficient of x^n in (1 + 5*x + 16*x^2)^n.

a(n) ~ 13^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 10 2018

D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) + 39*(n-1)*a(n-2) for n > 1. - Seiichi Manyama, Apr 22 2019

a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025