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%I #21 Aug 02 2021 06:33:14
%S 1,3,8,17,17,-78,-585,-2021,-1710,29395,231413,856264,-346979,
%T -30019585,-232782792,-834712259,2313820717,59793779314,469729578123,
%U 1597321309383,-9914171906614,-206169178856073,-1697255630380351,-5677886943413120,55801423903125353
%N Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).
%C In order to have A143496 as a lower triangular Sheffer matrix one uses row and column offsets 0 (not 4).
%H Seiichi Manyama, <a href="/A193684/b193684.txt">Table of n, a(n) for n = 0..591</a>
%F E.g.f.: exp(-exp(x)+4*x+1).
%F a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 4)^n / k!. - _Ilya Gutkovskiy_, Dec 20 2019
%F a(0) = 1; a(n) = 4 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - _Seiichi Manyama_, Aug 02 2021
%e With offset [0,0] row n=3 of A143496 is [64,61,15,1], hence a(3)=64-61+15-1=17.
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+4*x+1))) \\ _Michel Marcus_, Aug 02 2021
%Y Cf. A143496, A193683 (3-restricted Stirling2 case), A196835, A293037, A346739.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 06 2011