A386939 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(3*n-k-1,n-k).

1, 7, 82, 1083, 15086, 216566, 3169636, 47020371, 704497750, 10636206306, 161553957500, 2465911305182, 37791965926092, 581171323026508, 8963417696439752, 138590900605115779, 2147571141595692390, 33342454213792397930, 518548824827926272268, 8076888443386745743530

Offset: 0

Author

Seiichi Manyama, Aug 10 2025

Keywords

• nonn
• changed

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A386834, A386938.

Programs

Magma

[&+[Binomial(4*n+1,k) * Binomial(3*n-k-1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Mathematica

Table[Sum[Binomial[4*n+1, k]*Binomial[3*n-k-1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)

PARI

a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(3*n-k-1, n-k));

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(2*n).

a(n) = [x^n] 1/((1-x)^(n+2) * (1-2*x)^(2*n)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(2*n-k+1,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k-1,k) * binomial(2*n-k+1,n-k).