A390715 a(0) = 1; a(n) = 2 * Sum_{k=0..n} k * binomial(5*n-3*k,n-k)/(5*n-3*k).

1, 1, 3, 16, 113, 923, 8210, 77191, 754533, 7590798, 78076583, 817325680, 8679284686, 93266690776, 1012308668521, 11081799884030, 122212279519291, 1356497402257962, 15142172265100568, 169880925049047545, 1914497938726332443, 21663127366191475278, 246022093949813339604

Offset

0,3

Links

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

Formula

G.f.: 1/(1-x*g^2) where g = 1+x*g^5 is the g.f. of A002294.

a(n) = 2*binomial(5*n-3, n-1)*hypergeom([2, 1-2*n, (3-4*n)/2, 1-n], [(4-5*n)/3, 5*(1-n)/3, (6-5*n)/3], 2^2/3^3)/(5*n - 3) for n > 0. - Stefano Spezia, Nov 16 2025

Mathematica

a[n_]:=2*Binomial[5*n-3, n-1]*HypergeometricPFQ[{2, 1-2n, (3-4n)/2, 1-n}, {(4-5n)/3, 5(1-n)/3, (6-5n)/3}, 4/27]/(5n-3); Join[{1}, Array[a, 22]] (* Stefano Spezia, Nov 16 2025 *)
Join[{1}, Table[2*Sum[k*Binomial[5*n -3*k, n-k]/(5*n-3*k), {k, 0, n}], {n, 1, 25}]] (* Vincenzo Librandi, Nov 18 2025 *)

Prog

(PARI) a(n) = if(n==0, 1, 2*sum(k=0, n, k*binomial(5*n-3*k, n-k)/(5*n-3*k)));
(Magma) [1] cat [2*&+[k*Binomial(5*n-3*k, n-k)/(5*n-3*k): k in [0..n]] : n in [1..30] ]; // Vincenzo Librandi, Nov 18 2025

Crossrefs

Cf. A002294, A186182, A390716.

Sequence in context: A393738 A005419 A379193 * A381029 A332024 A124537

Adjacent sequences: A390712 A390713 A390714 * A390716 A390717 A390718

Keyword

nonn

Author

Seiichi Manyama, Nov 15 2025

Status

approved