A104454 Expansion of 1/(sqrt(1-5x)*sqrt(1-9x)).

1, 7, 51, 385, 2995, 23877, 194109, 1602447, 13389075, 112935445, 959783881, 8206116387, 70507643101, 608271899515, 5265458413875, 45711784088145, 397829544860115, 3469772959954245, 30319709631711225, 265383615634224675, 2326318766651511945, 20419439617056272415

Offset: 0

Author

Paul Barry, Mar 08 2005

Keywords

• easy
• nonn
• changed

Comments

Fifth binomial transform of A000984. In general, the k-th binomial transform of A000984 will have g.f. 1/(sqrt(1-k*x)*sqrt(1-(k+4)*x)) and a(n)=sum{i=0..n, C(n,i)C(2i,i)k^(n-i)}.

Diagonal of rational function 1/(1 - (x^2 + 7*x*y + y^2)). - Gheorghe Coserea, Aug 03 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Column 7 of A292627.

Cf. A081671, A098409, A098410.

Programs

Mathematica

CoefficientList[Series[1/(Sqrt[1-5x] Sqrt[1-9x]),{x,0,30}],x] (* Harvey P. Dale, Apr 11 2012 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-14*x+45*x^2)) \\ Joerg Arndt, May 13 2013

PARI

{a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

PARI

{a(n) = sum(k=0, n\2, 7^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

Formula

G.f.: 1/sqrt(1-14*x+45*x^2).

E.g.f.: exp(7x)*BesselI(0, 2x)

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

D-finite with recurrence: n*a(n) = 7*(2*n-1)*a(n-1) - 45*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 3^(2*n+1)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012

a(n) = Sum_{k=0..n} 9^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019

a(n) = Sum_{k=0..floor(n/2)} 7^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019

a(n) = (1/4)^n * Sum_{k=0..n} 5^k * 9^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025