A373433 a(n) = A000111(n) * A000142(n). Row sums of A373434.

1, 1, 2, 12, 120, 1920, 43920, 1370880, 55843200, 2879815680, 183330604800, 14122244505600, 1294628759424000, 139287595371724800, 17379949655535667200, 2489494639794978816000, 405724534220435189760000, 74646464089618378653696000, 15396938399483145082626048000

Offset: 0

Author

Peter Luschny, Jun 04 2024

Keywords

• nonn

Crossrefs

Cf. A000111, A000142, A373434.

Programs

Maple

A373433 := n -> ifelse(n = 0, 1, n! * 2^n * abs(euler(n, 1/2) + euler(n, 1))):
seq(A373433(n), n = 0..18);

Mathematica

A373433[n_] := 2 I^(n + 1) n! PolyLog[-n, -I]; A373433[0] := 1;
Table[A373433[n], {n, 0, 18}]

SageMath

# Algorithm of Ludwig Seidel (1877).
def A373433_list(n) :
    R = []; S = []; A = {-1:0, 0:1}; k = 0; f = 1; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        R.append(Am); S.append(f*Am); f *= i + 1
    return S
print(A373433_list(18))

Formula

a(n) = n! * 2^n * |Euler(n, 1/2) + Euler(n, 1)| for n >= 1.

a(n) ~ ((2*n^2)/(Pi*e^2))^n*(8*n + 4/3).