A248168 Expansion of g.f. 1/sqrt((1-3*x)*(1-11*x)).

1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, 518839783, 5414767897, 56795795679, 598213529809, 6322787125207, 67026654455433, 712352213507151, 7587639773475777, 80977812878889927, 865716569022673401, 9269461606674304959, 99387936492243451569, 1066975862517563301303

Offset: 0

Author

Paul D. Hanna, Oct 03 2014

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 7*x + 57*x^2 + 511*x^3 + 4849*x^4 + 47607*x^5 +...
where A(x)^2 = 1/((1-3*x)*(1-11*x)):
A(x)^2 = 1 + 14*x + 163*x^2 + 1820*x^3 + 20101*x^4 + 221354*x^5 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..961
• Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• 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. 97.

Crossrefs

Cf. A248167, A084771.

Programs

Magma

[n le 2 select 7^(n-1) else (7*(2*n-3)*Self(n-1) - 33*(n-2)*Self(n-2))/(n-1) : n in [1..40]]; // G. C. Greubel, May 31 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1-3*x)*(1-11*x)], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2014 *)

PARI

{a(n)=polcoeff( 1 / sqrt((1-3*x)*(1-11*x) +x*O(x^n)), n) }
for(n=0, 25, print1(a(n), ", "))

PARI

{a(n)=polcoeff( (1 + 7*x + 4*x^2 +x*O(x^n))^n, n) }
for(n=0, 25, print1(a(n), ", "))

PARI

{a(n)=sum(k=0,n, 3^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0, 25, print1(a(n), ", "))

SageMath

@CachedFunction
def A248168(n):
     if (n<2): return 7^n
     else: return (7*(2*n-1)*A248168(n-1) - 33*(n-1)*A248168(n-2))//n
print([A248168(n) for n in range(41)]) # G. C. Greubel, May 31 2025

Formula

a(n) equals the central coefficient in (1 + 7*x + 4*x^2)^n, n>=0.

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

a(n) = Sum_{k=0..n} 11^(n-k) * (-2)^k * C(n,k) * C(2*k,k). - Paul D. Hanna, Apr 20 2019

a(n)^2 = A248167(n), which gives the coefficients in 1 / AGM(1-3*11*x, sqrt((1-3^2*x)*(1-11^2*x))).

Equals the binomial transform of 2^n*A026375(n).

Equals the second binomial transform of A084771.

Equals the third binomial transform of A059304(n) = 2^n*(2*n)!/(n!)^2.

a(n) ~ 11^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 03 2014

D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +33*(n-1)*a(n-2)=0. [Belbachir]

a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 11^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025