A390713 a(0) = 1; a(n) = Sum_{k=0..n} k * binomial(4*n-2*k,n-k)/(2*n-k).

1, 1, 3, 14, 81, 528, 3706, 27343, 209091, 1642505, 13175592, 107475228, 888788286, 7434421879, 62790108909, 534724808122, 4586532562769, 39588016078536, 343595160001123, 2996862925155061, 26254154919371088, 230913398343788951, 2038243761068156956

Offset

0,3

Links

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

Formula

G.f.: 1/(1-x*g^2) where g = 1+x*g^4 is the g.f. of A002293.

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

Mathematica

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

Prog

(PARI) a(n) = if(n==0, 1, sum(k=0, n, k*binomial(4*n-2*k, n-k)/(2*n-k)));
(Magma) [1] cat [&+[k*Binomial(4*n-2*k, n-k)/(2*n-k): k in [0..n]] : n in [1..30] ]; // Vincenzo Librandi, Nov 18 2025

Crossrefs

Cf. A002293, A130458, A390714.

Sequence in context: A256336 A256338 A256331 * A292875 A077054 A355291

Adjacent sequences: A390710 A390711 A390712 * A390714 A390715 A390716

Keyword

nonn

Author

Seiichi Manyama, Nov 15 2025

Status

approved