A390714 a(0) = 1; a(n) = 5 * Sum_{k=0..n} k * binomial(4*n+k,n-k)/(4*n+k).

1, 1, 6, 41, 301, 2311, 18297, 148158, 1220534, 10193005, 86075121, 733601936, 6301301927, 54488332154, 473909547807, 4142807726591, 36378578332933, 320726976574879, 2837821013193630, 25190804896317777, 224272539715594201, 2002047393443251488, 17915898357703695344

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Formula

G.f.: 1/(1-x*g^5) where g = 1+x*g^4 is the g.f. of A002293.

a(n) = 5*binomial(4*n+1, n-1)*hypergeom([2, 1-n, 1+4*n], [3*(1+n)/2, (4+3*n)/2], -1/2^2)/(4*n + 1) for n > 0. - Stefano Spezia, Nov 16 2025

Mathematica

a[n_]:=5*Binomial[4*n+1, n-1]*HypergeometricPFQ[{2, 1-n, 1+4n}, {3(1+n)/2, (4+3n)/2}, -1/4]/(4n+1); Join[{1}, Array[a, 22]] (* Stefano Spezia, Nov 16 2025 *)
Join[{1}, Table[5*Sum[k*Binomial[4*n +k, n-k]/(4*n+k), {k, 0, n}], {n, 1, 25}]] (* Vincenzo Librandi, Nov 18 2025 *)

Prog

(PARI) a(n) = if(n==0, 1, 5*sum(k=0, n, k*binomial(4*n+k, n-k)/(4*n+k)));
(Magma) [1] cat [5*&+[k*Binomial(4*n+k, n-k)/(4*n+k): k in [0..n]] : n in [1..30] ]; // Vincenzo Librandi, Nov 18 2025

Crossrefs

Cf. A002293, A130458, A390713.

Sequence in context: A391494 A000402 A186654 * A152107 A143023 A078009

Adjacent sequences: A390711 A390712 A390713 * A390715 A390716 A390717

Keyword

nonn

Author

Seiichi Manyama, Nov 15 2025

Status

approved