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%I #15 Jan 22 2024 16:24:53
%S 0,0,0,3,6,20,180,1134,7980,78840,798840,8620920,107668440,1449377280,
%T 20755871136,323448048000,5398086002400,95487623038080,
%U 1796842848654720,35808112038746880,751616958775939200,16600116241063514880,384905905873078867200
%N Expansion of e.g.f. -log(1 + x^2/2 * log(1 - x)).
%F a(n) = n! * Sum_{k=1..floor(n/3)} (k-1)! * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
%t With[{nn =30},CoefficientList[Series[-Log[1+x^2/2 Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jan 22 2024 *)
%o (PARI) a(n) = n!*sum(k=1, n\3, (k-1)!*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
%Y Cf. A052804, A368166.
%Y Cf. A351505.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Dec 14 2023