A390888 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * Stirling2(n,k).

1, 2, 10, 62, 466, 4094, 40834, 453422, 5528530, 73245086, 1045617442, 15974829710, 259729905394, 4472698102526, 81249057380674, 1551476854161134, 31047024669646354, 649336273468833374, 14159652822131771746, 321245505385300938062, 7568059628217164709682

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

G.f.: Sum_{k>=0} (2*x)^k / Product_{j=1..k} (1 - 3*j*x).

E.g.f.: exp( 2*(exp(3*x)-1)/3 ).

a(n) = exp(-2/3) * Sum_{k>=0} 2^k * 3^(n-k) * k^n/k!.

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

Mathematica

Join[{1}, Table[Sum[2^k*3^(n-k)*StirlingS2[n, k], {k, 0, n}], {n, 25}]] (* Vincenzo Librandi, Jan 04 2026 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*3^(n-k)*stirling(n, k, 2));
(Magma) [1] cat [&+[ 2^k*3^(n-k)*StirlingSecond(n, k): k in [0..n]]: n in [1..25]]; // Vincenzo Librandi, Jan 04 2026

Crossrefs

Cf. A004212, A247452, A390889.

Cf. A008544.

Sequence in context: A192942 A307364 A141140 * A232472 A380558 A361829

Adjacent sequences: A390885 A390886 A390887 * A390889 A390890 A390891

Keyword

nonn

Author

Seiichi Manyama, Nov 22 2025

Status

approved