A263575 Stirling transform of Lucas numbers (A000032).

2, 1, 4, 14, 53, 227, 1092, 5791, 33350, 206511, 1365563, 9590847, 71216713, 556861216, 4569168866, 39222394456, 351304769679, 3275433717440, 31723522878974, 318571978752719, 3311400814816987, 35573458376435132, 394404160256111139, 4507130777468928696

Offset

0,1

Links

Alois P. Heinz, Table of n, a(n) for n = 0..563

Eric Weisstein's MathWorld, Lucas Number.

Eric Weisstein's MathWorld, Stirling Transform.

Eric Weisstein's MathWorld, Bell Polynomial.

Formula

a(n) = Sum_{k=0..n} A000032(k)*Stirling2(n,k).

Let phi = (1+sqrt(5))/2.

a(n) = B_n(phi)+B_n(1-phi), where B_n(x) is n-th Bell polynomial.

2*B_n(phi) = a(n) + A263576*sqrt(5).

E.g.f.: exp((exp(x)-1)*phi)+exp((exp(x)-1)*(1-phi)).

Sum_{k=0..n} a(k)*Stirling1(n,k) = A000032(n).

G.f.: Sum_{j>=0} Lucas(j)*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019

Mathematica

Table[Sum[LucasL[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
Table[Simplify[BellB[n, GoldenRatio] + BellB[n, 1 - GoldenRatio]], {n, 0, 23}]

Crossrefs

Cf. A000032, A213593, A005248, A061084, A263576.

Sequence in context: A102055 A232376 A350739 * A162977 A032174 A212267

Adjacent sequences: A263572 A263573 A263574 * A263576 A263577 A263578

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 21 2015

Status

approved