A387037 a(n) = Sum_{k=0..n} binomial(4*n-1,k).
1, 4, 29, 232, 1941, 16664, 145499, 1285624, 11460949, 102875128, 928495764, 8417689504, 76599066579, 699232769512, 6400175653922, 58718827590992, 539822826733397, 4971747032359352, 45863130731297180, 423683961417124576, 3919058645835901556
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Magma
[&+[Binomial(4*n-1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
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Mathematica
Table[Sum[Binomial[4*n-1,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n-1, k));
Formula
a(n) = [x^n] (1+x)^(4*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-1,k) * binomial(4*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-2,n-k).
G.f.: 1/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-5)*(4*n-7)*(2*n-3)*(11*n^2-3*n-3)*a(n-2) -8*(946*n^5-4218*n^4+6512*n^3-3753*n^2+201*n+315)*a(n-1) +3*n*(3*n-2)*(3*n-4)*(11*n^2-25*n+11)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi*n) * 3^(3*n - 3/2)). - Vaclav Kotesovec, Sep 03 2025