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A049402 Row sums of triangle A049374.

%I A049402 #17 Jan 18 2025 09:05:40
%S A049402 1,1,7,61,649,8245,122215,2069425,39328465,827226505,19047582055,
%T A049402 475956135205,12815133759385,369605936607805,11361372997850695,
%U A049402 370609338222772825,12780705695068446625,464412124831585889425,17729002673226394402375,709180766131239680070925
%N A049402 Row sums of triangle A049374.
%H A049402 Alois P. Heinz, <a href="/A049402/b049402.txt">Table of n, a(n) for n = 0..415</a>
%H A049402 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 A049402 E.g.f. exp(p(x)) with p(x) := (1-(1-x)^5)/(5*(1-x)^5) (E.g.f. first column of A049374).
%F A049402 From _Seiichi Manyama_, Jan 18 2025: (Start)
%F A049402 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005011(k).
%F A049402 a(n) = (1/exp(1/5)) * (-1)^n * n! * Sum_{k>=0} binomial(-5*k,n)/(5^k * k!). (End)
%p A049402 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A049402       binomial(n-1, j-1)*(j+4)!/5!*a(n-j), j=1..n))
%p A049402     end:
%p A049402 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 01 2017
%t A049402 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 4)!/5!*a[n - j], {j, 1, n}];
%t A049402 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A049402 Column k=6 of A291709.
%Y A049402 Cf. A049374, A049427.
%K A049402 easy,nonn
%O A049402 0,3
%A A049402 _Wolfdieter Lang_
%E A049402 a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2017