A319177 -id:A319177 - cp's OEIS frontend

A319175 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k!)^n.

1, 1, 4, 36, 416, 6000, 106542, 2242093, 54399424, 1495318752, 45938780750, 1559858659359, 58007497143180, 2344682328265823, 102352889947823998, 4798930456964580045, 240518006611511552896, 12832137350594892322464, 726108032647676403262710, 43434461707962856186584307

Offset: 0

Ilya Gutkovskiy, Sep 12 2018

nonn

Cf. A007837, A032312, A032314, A032315, A319174, A319177.

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)).

A319176 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k)^n.

Original entry on oeis.org

1, 1, 8, 93, 1532, 32240, 829284, 25192454, 882825936, 35055329832, 1555548490560, 76285107738312, 4097094075364608, 239167754501235456, 15077741379436233120, 1020918130521930465120, 73892194568147257761024, 5693112248722998479169408, 465208700406183224884224000

Offset: 0

Ilya Gutkovskiy, Sep 12 2018

nonn

Cf. A007841, A294469, A299034, A319174, A319177.

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)).

cp's OEIS Frontend

A319175 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k!)^n.

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