A242586 Expansion of 1/(2*sqrt(1-x))*(1/sqrt(1-x)+1/(sqrt(1-5*x))).
1, 2, 6, 23, 98, 437, 1995, 9242, 43258, 204053, 968441, 4619012, 22120631, 106300508, 512321438, 2475395303, 11986728458, 58156146653, 282640193313, 1375737276788, 6705522150973, 32724071280518, 159878425878848
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
Programs
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Maple
a := n -> hypergeom([1/2,-n],[1], -4)/2 + 1/2; seq(round(evalf(a(n),32)), n=0..22); # Peter Luschny, May 18 2014
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Mathematica
CoefficientList[Series[1/(2Sqrt[1-x]) (1/Sqrt[1-x]+1/Sqrt[1-5x]),{x,0,30}],x] (* Harvey P. Dale, Mar 19 2020 *) -
Maxima
a(n):=sum(binomial(2*j-1,j)*binomial(n,j),j,0,n);
Formula
a(n) = Sum_{j = 0..n} binomial(2*j-1,j)*binomial(n,j).
G.f. A(x) = x*F'(x)/F(x), where F(x) is g.f. of A007317.
a(n) = T(2*n,n) for n>0, where T(n,k) is triangle of A105477.
a(n) = hypergeom([1/2,-n],[1],-4)/2 + 1/2. - Peter Luschny, May 18 2014
D-finite with recurrence: n*a(n) + (-7*n+4)*a(n-1) + (11*n-14)*a(n-2) + 5*(-n+2)*a(n-3) = 0. - R. J. Mathar, May 23 2014
2*a(n) = 1 + A026375(n). - R. J. Mathar, Jan 26 2020
From Peter Bala, Jan 09 2022: (Start)
a(n) = [x^n] ( x + 1/(1 - x) )^n.
a(0) = 1, a(1) = 2 and n*a(n) = 3*(2*n-1)*a(n-1) - 5*(n-1)*a(n-2) - 1 for n >= 2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
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