A104454 - OEIS

%I #42 Jan 30 2020 21:29:15

%S 1,7,51,385,2995,23877,194109,1602447,13389075,112935445,959783881,

%T 8206116387,70507643101,608271899515,5265458413875,45711784088145,

%U 397829544860115,3469772959954245,30319709631711225,265383615634224675,2326318766651511945,20419439617056272415

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

%C 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)}.

%C Diagonal of rational function 1/(1 - (x^2 + 7*x*y + y^2)). - _Gheorghe Coserea_, Aug 03 2018

%H Seiichi Manyama, <a href="/A104454/b104454.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)

%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

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

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

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

%F 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

%F a(n) ~ 3^(2*n+1)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 17 2012

%F a(n) = Sum_{k=0..n} 9^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - _Seiichi Manyama_, Apr 22 2019

%F 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

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

%o (PARI) x='x+O('x^66); Vec(1/sqrt(1-14*x+45*x^2)) \\ _Joerg Arndt_, May 13 2013

%o (PARI) {a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ _Seiichi Manyama_, Apr 22 2019

%o (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

%Y Column 7 of A292627.

%Y Cf. A081671, A098409, A098410.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 08 2005