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A049378 Row sums of triangle A049353.

%I A049378 #17 Jan 18 2025 09:05:36
%S A049378 1,1,6,46,436,4956,65776,996976,16957536,319259296,6581662336,
%T A049378 147290942976,3552885191296,91827536814976,2530228890080256,
%U A049378 74003737259670016,2288810287491774976,74607500831801289216,2555587654482227055616,91746983502042106018816
%N A049378 Row sums of triangle A049353.
%H A049378 Alois P. Heinz, <a href="/A049378/b049378.txt">Table of n, a(n) for n = 0..421</a>
%H A049378 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 A049378 E.g.f. exp(p(x)) with p(x) := x*(2-x)*(2-2*x+x^2)/(4*(1-x)^4) (E.g.f. first column of A049353).
%F A049378 From _Seiichi Manyama_, Jan 18 2025: (Start)
%F A049378 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004213(k).
%F A049378 a(n) = (1/exp(1/4)) * (-1)^n * n! * Sum_{k>=0} binomial(-4*k,n)/(4^k * k!). (End)
%p A049378 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A049378       binomial(n-1, j-1)*(j+3)!/4!*a(n-j), j=1..n))
%p A049378     end:
%p A049378 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 01 2017
%t A049378 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 3)!/4!*a[n - j], {j, 1, n}];
%t A049378 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A049378 Column k=5 of A291709.
%Y A049378 Cf. A049353, A049426.
%K A049378 nonn
%O A049378 0,3
%A A049378 _Wolfdieter Lang_
%E A049378 a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2017