This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263575 #18 Feb 16 2025 08:33:27
%S A263575 2,1,4,14,53,227,1092,5791,33350,206511,1365563,9590847,71216713,
%T A263575 556861216,4569168866,39222394456,351304769679,3275433717440,
%U A263575 31723522878974,318571978752719,3311400814816987,35573458376435132,394404160256111139,4507130777468928696
%N A263575 Stirling transform of Lucas numbers (A000032).
%H A263575 Alois P. Heinz, <a href="/A263575/b263575.txt">Table of n, a(n) for n = 0..563</a>
%H A263575 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%H A263575 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>.
%H A263575 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A263575 a(n) = Sum_{k=0..n} A000032(k)*Stirling2(n,k).
%F A263575 Let phi = (1+sqrt(5))/2.
%F A263575 a(n) = B_n(phi)+B_n(1-phi), where B_n(x) is n-th Bell polynomial.
%F A263575 2*B_n(phi) = a(n) + A263576*sqrt(5).
%F A263575 E.g.f.: exp((exp(x)-1)*phi)+exp((exp(x)-1)*(1-phi)).
%F A263575 Sum_{k=0..n} a(k)*Stirling1(n,k) = A000032(n).
%F A263575 G.f.: Sum_{j>=0} Lucas(j)*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 06 2019
%t A263575 Table[Sum[LucasL[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
%t A263575 Table[Simplify[BellB[n, GoldenRatio] + BellB[n, 1 - GoldenRatio]], {n, 0, 23}]
%Y A263575 Cf. A000032, A213593, A005248, A061084, A263576.
%K A263575 nonn
%O A263575 0,1
%A A263575 _Vladimir Reshetnikov_, Oct 21 2015