A307375 Expansion of Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k^2*x).

1, 1, 3, 17, 151, 1893, 31499, 666169, 17351967, 543441005, 20079329875, 861908850561, 42439075349543, 2371469004695797, 149022897087857691, 10448429535366899273, 811758520658841809839, 69463012765807086749949, 6511800419610377560644707, 665560984365147223546851985

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2019

Keywords

• nonn

Comments

a(n) is the number of partitions of [2n] such that the largest element of each block is even. a(3) = 17: 123456, 1234|56, 12356|4, 124|356, 1256|34, 12|3456, 12|34|56, 12|356|4, 13456|2, 134|256, 134|2|56, 1356|24, 1356|2|4, 14|2356, 156|234, 14|2|356, 156|2|34. - Alois P. Heinz, Jun 10 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..303

Crossrefs

Cf. A000670, A135920, A229234, A363589.

Bisection of A290383 (even part).

Programs

Maple

b:= proc(n, x, y) option remember; `if`(n=0, 1, `if`(n::odd, 0,
      b(n-1, y, x+1))+b(n-1, y, x)*x+b(n-1, y, x)*y)
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..19);  # Alois P. Heinz, Jun 10 2023

Mathematica

nmax = 19; CoefficientList[Series[Sum[j! x^j/Product[(1 - k^2 x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]