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A032033 Stirling transform of A032031.

%I A032033 #69 Jun 02 2025 05:26:49
%S A032033 1,3,21,219,3045,52923,1103781,26857659,746870565,23365498683,
%T A032033 812198635941,31055758599099,1295419975298085,58538439796931643,
%U A032033 2848763394161128101,148537065755389540539,8261178848690959117605,488177936257344615487803,30544839926043868901604261
%N A032033 Stirling transform of A032031.
%C A032033 Also "AIJ" (ordered, indistinct, labeled) transform of 3,3,3,3...
%C A032033 Third row of array A094416 (generalized ordered Bell numbers).
%H A032033 Vincenzo Librandi, <a href="/A032033/b032033.txt">Table of n, a(n) for n = 0..200</a>
%H A032033 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A032033 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0303030">Dobinski-type relations and the log-normal distribution</a>, arXiv:quant-ph/0303030, J. Phys. A.: Math. Gen  36 (2003) L273.
%H A032033 Christian G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032033 Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%F A032033 E.g.f.: 1/(4-3*exp(x)).
%F A032033 a(n) = 3 * A050352(n), n > 0.
%F A032033 a(n) = Sum_{k=0..n} Stirling2(n, k) * (3^k) * k!.
%F A032033 a(n) = (1/4) * Sum_{k>=0} k^n * (3/4)^k. - _Karol A. Penson_, Jan 25 2002
%F A032033 a(n) = Sum_{k=0..n} A131689(n,k)*3^k. - _Philippe Deléham_, Nov 03 2008
%F A032033 G.f. A(x)=B(x)/x, where B(x)=x+3*x^2+21*x^3+... = Sum_{n>=1} b(n)*x^n satisfies 4*B(x)-x = 3*B(x/(1-x)), and b(n)=3*Sum_{k=1..n-1} binomial(n-1,k-1)*b(k), b(1)=1. - _Vladimir Kruchinin_, Jan 27 2011
%F A032033 a(n) = log(4/3)*Integral_{x = 0..inf} (floor(x))^n * (4/3)^(-x) dx. - _Peter Bala_, Feb 14 2015
%F A032033 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020
%F A032033 a(0) = 1; a(n) = 3 * a(n-1) - 4 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023
%F A032033 a(n) = (3/4) * Sum_{k=0..n} 4^k * (-1)^(n-k) * k! * Stirling2(n,k) for n > 0. - _Seiichi Manyama_, Jun 01 2025
%p A032033 b:= proc(n, m) option remember;
%p A032033      `if`(n=0, 3^m*m!, m*b(n-1, m)+b(n-1, m+1))
%p A032033     end:
%p A032033 a:= n-> b(n, 0):
%p A032033 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2021
%t A032033 a[n_] := PolyLog[-n, 3/4]/4; a[0] = 1; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Nov 14 2011 *)
%t A032033 t = 30; Range[0, t]! CoefficientList[Series[1/(4 - 3 Exp[x]), {x, 0, t}], x] (* _Vincenzo Librandi_, Mar 16 2014 *)
%o A032033 (PARI) a(n)=ceil(polylog(-n,3/4)/4) \\ _Charles R Greathouse IV_, Jul 14 2014
%o A032033 (PARI) my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(4 - 3*exp(x)))) \\ _Joerg Arndt_, Jan 15 2024
%Y A032033 Cf. A032031, A094416, A094418, A094419.
%K A032033 nonn,easy
%O A032033 0,2
%A A032033 _Christian G. Bower_