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%I #37 Aug 02 2021 02:31:08
%S 1,2,3,1,-14,-59,-99,288,2885,10365,1700,-226313,-1535203,-4258630,
%T 17243695,284513877,1688253890,2750940953,-51540956455,-624352447488,
%U -3470378651847,-496964048927,204678286709292,2311290490508227,12611758414937801
%N Alternating row sums of Sheffer triangle A143495 (3-restricted Stirling2 numbers).
%C In order to have a lower triangular Sheffer matrix for A143495 one uses row and column offsets 0 (not 3).
%D See A143495.
%H Seiichi Manyama, <a href="/A193683/b193683.txt">Table of n, a(n) for n = 0..591</a>
%F E.g.f.: exp(-exp(x)+3*x+1).
%F G.f.: (1 - 2/E(0))/x where E(k) = 1 + 1/(1 - 2*x/(1 - 2*(k+1)*x/E(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Sep 20 2012
%F G.f.: 1/U(0) where U(k) = 1 - x*(k+2) + x^2*(k+1)/U(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 11 2012
%F G.f.: (1 - G(0) )/(x+1) where G(k) = 1 - 1/(1-k*x-3*x)/(1-x/(x+1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 17 2013
%F G.f.: 1/Q(0), where Q(k) = 1 - 3*x + x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 22 2013
%F G.f.: G(0)/(1-2*x), where G(k) = 1 - x^2*(2*k+1)/(x^2*(2*k+1) + (1-x*(2*k+2))*(1-x*(2*k+3))/(1 - x^2*(2*k+2)/(x^2*(2*k+2) + (1-x*(2*k+3))*(1-x*(2*k+4))/G(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Feb 06 2014
%F a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 3)^n / k!. - _Ilya Gutkovskiy_, Dec 20 2019
%F a(0) = 1; a(n) = 3 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - _Seiichi Manyama_, Aug 02 2021
%e Row no. 3 of A143495 with [0,0] offset is [27,37,12,1], hence a(3)=27-37+12-1=1.
%t With[{nn=30},CoefficientList[Series[Exp[3x+1-Exp[x]],{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Jan 10 2013 *)
%Y Cf. A143495, A074051 (2-restricted Stirling2 case), A193684, A196835, A293037, A346738.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 06 2011
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