A319176 -id:A319176 - cp's OEIS frontend

A319174 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k!)^n.

1, 1, 8, 90, 1448, 29750, 747462, 22182741, 759504720, 29468021238, 1277744462870, 61232148035531, 3213710056592796, 183329936018667035, 11294683874759287030, 747379761629288205795, 52864744954736491460768, 3980505280416276751035270, 317877846102688099315299678

Offset: 0

Ilya Gutkovskiy, Sep 12 2018

nonn

Cf. A005651, A174661, A319175, A319176.

Table[n! SeriesCoefficient[Product[1/(1 - x^k/k!)^n, {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[x^(j k)/(k (j!)^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 18}]

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)).

A319177 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k)^n.

Original entry on oeis.org

1, 1, 4, 39, 500, 7990, 156684, 3640392, 97543088, 2960758800, 100428661440, 3764849536800, 154567280328768, 6897265807262064, 332386213584653760, 17204016957686536320, 951852354201532742400, 56059949872552858763520, 3501729575599545174352896, 231227806715994322631352960

Offset: 0

Ilya Gutkovskiy, Sep 12 2018

nonn

Cf. A007838, A181541, A300187, A319175, A319176.

Table[n! SeriesCoefficient[Product[(1 + x^k/k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[(-1)^(k + 1) x^(j k)/(k j^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 19}]

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*j^k)).

cp's OEIS Frontend

A319174 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k!)^n.

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