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A104454 Expansion of 1/(sqrt(1-5x)*sqrt(1-9x)). 8
1, 7, 51, 385, 2995, 23877, 194109, 1602447, 13389075, 112935445, 959783881, 8206116387, 70507643101, 608271899515, 5265458413875, 45711784088145, 397829544860115, 3469772959954245, 30319709631711225, 265383615634224675, 2326318766651511945, 20419439617056272415 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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: 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

MATHEMATICA

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

PROG

(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

CROSSREFS

Column 7 of A292627.

Cf. A081671, A098409, A098410.

Sequence in context: A162757 A285880 A147958 * A222849 A273055 A019472

Adjacent sequences:  A104451 A104452 A104453 * A104455 A104456 A104457

KEYWORD

easy,nonn,changed

AUTHOR

Paul Barry, Mar 08 2005

STATUS

approved

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Last modified January 20 05:45 EST 2020. Contains 331067 sequences. (Running on oeis4.)