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%I #20 Nov 29 2023 13:06:04
%S 1,-2,-5,19,178,175,-7739,-72056,-33179,6899311,87861076,215532301,
%T -11151014291,-222077806202,-1563185592617,22953386817343,
%U 878911293113026,12330887396253691,1416506544326449,-4284948239134152536
%N Alternating row sums of the Sheffer triangle (exp(x), exp(3*x) - 1) given in A282629.
%C See A282629 for details. This is a generalization of A000587.
%F a(n) = Sum_{m=0..n} (-1)^m*A282629(n, m), n >= 0.
%F E.g.f.: exp(x)*exp(1 - exp(3*x)).
%F a(n) = (1/e)*Sum_{m>=0} ((-1)^m / m!)*(1 + 3*m)^n, n >= 0, (Dobiński type formula).- _Wolfdieter Lang_, Apr 10 2017
%F a(0) = 1; a(n) = a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - _Ilya Gutkovskiy_, Nov 29 2023
%t Fold[#2 - #1 &, Reverse@ #] & /@ Table[Sum[Binomial[m, k] (-1)^(k - m) (1 + 3 k)^n/m!, {k, 0, m}], {n, 0, 19}, {m, 0, n}] (* _Michael De Vlieger_, Apr 08 2017 *)
%o (PARI) T(n, m) = sum(k=0, m, binomial(m, k) * (-1)^(k - m) * (1 + 3*k)^n/m!);
%o a(n) = sum(m=0, n, (-1)^m*T(n, m)); \\ _Indranil Ghosh_, Apr 10 2017
%Y Cf. A000110, A282629, A284859.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Apr 05 2017
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