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%I #14 Jun 23 2023 17:57:49
%S 0,1,3,12,57,305,1798,11531,79707,589426,4634471,38547861,337734048,
%T 3105588629,29877483743,299906019892,3133423928557,34002824654365,
%U 382507638525838,4452923233600903,53561431659306039,664728428775177890,8500763141347126563,111886109022440334593,1513989730079050155936
%N Stirling transform of Pell numbers (A000129).
%H Alois P. Heinz, <a href="/A323631/b323631.txt">Table of n, a(n) for n = 0..553</a>
%F E.g.f.: exp(exp(x) - 1)*sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
%F a(n) = Sum_{k=0..n} Stirling2(n,k)*A000129(k).
%F a(n) = Sum_{k=0..n} binomial(n,k)*A000110(n-k)*A264037(k).
%p b:= proc(n, m) option remember; `if`(n=0,
%p (<<2|1>, <1|0>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..24); # _Alois P. Heinz_, Jun 23 2023
%t FullSimplify[nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] Sinh[Sqrt[2] (Exp[x] - 1)]/Sqrt[2], {x, 0, nmax}], x] Range[0, nmax]!]
%t Table[Sum[StirlingS2[n, k] Fibonacci[k, 2], {k, 0, n}], {n, 0, 24}]
%t 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}]
%Y Cf. A000110, A000129, A263575, A263576, A264037.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 21 2019
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