A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).
1, 9, 79, 697, 6196, 55455, 499178, 4514873, 40999516, 373585604, 3414035527, 31278197839, 287191809724, 2642070371194, 24347999094724, 224723513577529, 2076978797223820, 19220104372823340, 178061257422521452, 1651314042800498052, 15328459501269535952
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
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Magma
[&+[Binomial(4*n+4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025
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Mathematica
Table[Sum[Binomial[4*n+4,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(4*n+4, k));
Formula
a(n) = [x^n] (1+x)^(4*n+4)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n+4) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+4,k) * binomial(4*n-k+3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+3,n-k).
G.f.: g^5/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-3)*(2*n+1)*(4*n-1)*(22*n^2+16*n-3)*a(n-2) -8*(1892*n^5+1024*n^4-1982*n^3-1306*n^2-60*n+27)*a(n-1) +3*(n+1)*(3*n+2)*(3*n+1)*(22*n^2-28*n+3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n + 15/2) / (sqrt(Pi*n) * 3^(3*n + 7/2)). - Vaclav Kotesovec, Aug 18 2025