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
Keywords
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.
Programs
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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