A336242 a(n) = (n!)^2 * Sum_{d|n} (-1)^(d+1) / (d!)^2.

1, 3, 37, 431, 14401, 403199, 25401601, 1216454399, 135339724801, 9877056537599, 1593350922240001, 178056522962841599, 38775788043632640001, 5700041141609893478399, 1757631343928533032960001, 327562346808114783805439999, 126513546505547170185216000001

Offset: 1

Author

Ilya Gutkovskiy, Jul 13 2020

Keywords

• nonn

Crossrefs

Cf. A073701, A132958, A336241.

Programs

Mathematica

Table[(n!)^2 Sum[(-1)^(d + 1)/(d!)^2, {d, Divisors[n]}], {n, 1, 17}]
nmax = 17; CoefficientList[Series[Sum[(1 - BesselJ[0, 2 x^(k/2)]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2 // Rest

PARI

a(n) = n!^2*sumdiv(n, d, (-1)^(d+1)/d!^2); \\ Michel Marcus, Jul 13 2020

Formula

a(n) = (n!)^2 * [x^n] Sum_{k>=1} (1 - BesselJ(0,2*x^(k/2))).

a(n) = (n!)^2 * [x^n] Sum_{k>=1} -(-x)^k / ((k!)^2 * (1 - x^k)).