A066381 a(n) = Sum_{k=0..n} binomial(4*n,k).
1, 5, 37, 299, 2517, 21700, 190051, 1683218, 15033173, 135142796, 1221246132, 11083374659, 100946732307, 922205369324, 8446802334994, 77542088287444, 713250450657109, 6572130378649468, 60652194138406780, 560522209086365852
Offset: 0
Keywords
Links
- Harry J. Smith, Table of n, a(n) for n = 0..150
Programs
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Maple
ogf := eval(s/((s-2)*(3*s-4)), s = RootOf(1-s+x*s^4, s)); series(ogf, x=0, 25); # Mark van Hoeij, May 05 2013
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Mathematica
Table[Sum[Binomial[4*n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 03 2015 *) -
Maxima
a[0]:1$ a[1]:5$ a[n]:=8*((3784*n^6-18764*n^5+34432*n^4 -28138*n^3+9028*n^2-24*n-315)*a[n-1]+16*(3-2*n)*(4*n-5)*(4*n-7)*(44*n^3-34*n^2-2*n+3)*a[n-2])/(3*n*(3*n-1)*(3*n-2)*(44*n^3-166*n^2 +198*n-73))$ makelist(a[n],n,0,1000); /* Tani Akinari, Sep 02 2014 */
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PARI
{ for (n=0, 150, a=0; for (k=0, n, a+=binomial(4*n, k)); write("b066381.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010
Formula
G.f.: s/((s-2)*(3*s-4)) where s = o.g.f.(A002293) which satisfies 1-s+x*s^4 = 0. - Mark van Hoeij, May 05 2013
a(0) = 1, a(n) = 16*a(n-1)-2*(44*n^3-34*n^2-2*n+3)*(4*n-4)!/(n!*(3*n-1)!). - Tani Akinari, Sep 02 2014
a(n) are special values of the hypergeometric function 2F1, in Maple notation: a(n)=16^n-binomial(4*n,n+1)*hypergeom([1,-3*n+1],[n+2],-1), n=0,1,... . - Karol A. Penson, Jun 03 2015
a(n) ~ (256/27)^n * sqrt(3/(2*Pi*n)). - Vaclav Kotesovec, Jun 03 2015
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(3*n)). - Ilya Gutkovskiy, Oct 25 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(4*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
G.f.: 1/(1 - x*g^3*(8-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025