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A049377 Row sums of triangle A049352.

%I A049377 #24 Jan 18 2025 09:05:32
%S A049377 1,1,5,33,273,2721,31701,421905,6302913,104270913,1889862021,
%T A049377 37204038081,789866524305,17977594555233,436435929785493,
%U A049377 11251798888929201,306889765901872641,8825681949708120705,266828094135981378693,8458295877281844310113
%N A049377 Row sums of triangle A049352.
%H A049377 Alois P. Heinz, <a href="/A049377/b049377.txt">Table of n, a(n) for n = 0..428</a>
%H A049377 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A049377 E.g.f. exp(p(x)) with p(x) := x*(3-3*x+x^2)/(3*(1-x)^3) (E.g.f. first column of A049352).
%F A049377 a(n) ~ n^(n-1/8)/2 * exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n). - _Vaclav Kotesovec_, Oct 23 2017
%F A049377 E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^4 dx )^n / n!, where the constant of integration is taken to be zero. - _Paul D. Hanna_, Apr 27 2019
%F A049377 From _Seiichi Manyama_, Jan 18 2025: (Start)
%F A049377 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004212(k).
%F A049377 a(n) = (1/exp(1/3)) * (-1)^n * n! * Sum_{k>=0} binomial(-3*k,n)/(3^k * k!). (End)
%p A049377 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A049377       binomial(n-1, j-1)*(j+2)!/6*a(n-j), j=1..n))
%p A049377     end:
%p A049377 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 01 2017
%t A049377 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*(j+2)!/6*a[n-j], {j, 1, n}];
%t A049377 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A049377 Column k=4 of A291709.
%Y A049377 Cf. A049352, A049425.
%K A049377 nonn
%O A049377 0,3
%A A049377 _Wolfdieter Lang_
%E A049377 a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2017