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%I #7 Jun 06 2024 04:28:23
%S 1,1,2,12,120,1920,43920,1370880,55843200,2879815680,183330604800,
%T 14122244505600,1294628759424000,139287595371724800,
%U 17379949655535667200,2489494639794978816000,405724534220435189760000,74646464089618378653696000,15396938399483145082626048000
%N a(n) = A000111(n) * A000142(n). Row sums of A373434.
%F a(n) = n! * 2^n * |Euler(n, 1/2) + Euler(n, 1)| for n >= 1.
%F a(n) ~ ((2*n^2)/(Pi*e^2))^n*(8*n + 4/3).
%p A373433 := n -> ifelse(n = 0, 1, n! * 2^n * abs(euler(n, 1/2) + euler(n, 1))):
%p seq(A373433(n), n = 0..18);
%t A373433[n_] := 2 I^(n + 1) n! PolyLog[-n, -I]; A373433[0] := 1;
%t Table[A373433[n], {n, 0, 18}]
%o (SageMath) # Algorithm of Ludwig Seidel (1877).
%o def A373433_list(n) :
%o R = []; S = []; A = {-1:0, 0:1}; k = 0; f = 1; e = 1
%o for i in (0..n) :
%o Am = 0; A[k + e] = 0; e = -e
%o for j in (0..i) : Am += A[k]; A[k] = Am; k += e
%o R.append(Am); S.append(f*Am); f *= i + 1
%o return S
%o print(A373433_list(18))
%Y Cf. A000111, A000142, A373434.
%K nonn
%O 0,3
%A _Peter Luschny_, Jun 04 2024
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