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

Author

Vladimir Reshetnikov, Oct 21 2015

Keywords

• nonn

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.

Crossrefs

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

Programs

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}]

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