A323631 Stirling transform of Pell numbers (A000129).

0, 1, 3, 12, 57, 305, 1798, 11531, 79707, 589426, 4634471, 38547861, 337734048, 3105588629, 29877483743, 299906019892, 3133423928557, 34002824654365, 382507638525838, 4452923233600903, 53561431659306039, 664728428775177890, 8500763141347126563, 111886109022440334593, 1513989730079050155936

Offset: 0

Author

Ilya Gutkovskiy, Jan 21 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..553

Crossrefs

Cf. A000110, A000129, A263575, A263576, A264037.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0,
      (<<2|1>, <1|0>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 23 2023

Mathematica

FullSimplify[nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] Sinh[Sqrt[2] (Exp[x] - 1)]/Sqrt[2], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS2[n, k] Fibonacci[k, 2], {k, 0, n}], {n, 0, 24}]
Table[Sum[Binomial[n, k] BellB[n - k] (BellB[k, Sqrt[2]] - BellB[k, -Sqrt[2]])/(2 Sqrt[2]), {k, 0, n}], {n, 0, 24}]

Formula

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

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000129(k).

a(n) = Sum_{k=0..n} binomial(n,k)*A000110(n-k)*A264037(k).