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A130410 Alternating row sums of triangle A130191 (Stirling2)^2.

%I A130410 #21 Feb 16 2025 08:33:06
%S A130410 1,-1,-1,0,6,32,115,172,-2030,-29013,-250051,-1587556,-5178877,
%T A130410 52922256,1435509569,20813187553,230664704969,1884809758791,
%U A130410 5120430335582,-216605840330716,-6440821191934686,-122368984222010397,-1842986108839510180,-21473141673616814694
%N A130410 Alternating row sums of triangle A130191 (Stirling2)^2.
%C A130410 Stirling2 transform of A000587. 2nd Stirling2 transform of A033999. - _Vladimir Reshetnikov_, Oct 22 2015
%H A130410 Robert Israel, <a href="/A130410/b130410.txt">Table of n, a(n) for n = 0..470</a>
%H A130410 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>.
%H A130410 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A130410 a(n) = sum(A130191(n,m)*(-1)^m,m=0..n), n>=0.
%F A130410 E.g.f.: 1/exp(f(x)) with f(x):=exp(exp(x)-1)-1.
%F A130410 a(n) = sum(k=0..n, A000587(k)*stirling2(n,k)) = sum(k=0..n, B_k(-1)*stirling2(n,k)), where B_k(x) is k-th Bell polynomial.
%e A130410 E.g.f.: 1 - x - (1/2)*x^2 + (1/4)*x^4+(4/15)*x^5 + (23/144)*x^6 + (43/1260)*x^7 - (29/576)*x^8 - (9671/120960)*x^9 ...
%e A130410 G.f. = 1 - x - x^2 + 6*x^4 + 32*x^5 + 115*x^6 + 172*x^7 - 2030*x^8 - 29013*x^9 + ...
%p A130410 Egf:= 1/exp(exp(exp(x)-1)-1):
%p A130410 S:= series(Egf,x,101):
%p A130410 seq(coeff(S,x,j)*j!, j=0..100); # _Robert Israel_, Oct 22 2015
%t A130410 Table[Sum[BellY[n, k, -BellB[Range[n]]], {k, 0, n}], {n, 0, 23}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%Y A130410 Cf. A048993, A000258 (row sums of A130191), A000587, A033999, A130191.
%K A130410 sign,easy
%O A130410 0,5
%A A130410 _Wolfdieter Lang_ Jun 01 2007