A319113 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k)/prime(k)).

1, 0, 1, 2, 0, 44, 0, 1224, 2688, 25920, 293760, 3628800, 25090560, 762048000, 3887170560, 62749209600, 1233908121600, 22616539545600, 321930878976000, 10717413809356800, 108951843667968000, 1982497256570880000, 50138292140310528000, 1408088823809310720000, 25175914255793258496000

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• nonn

Crossrefs

Cf. A002099, A007838, A319112.

Programs

Maple

seq(n!*coeff(series(mul((1+x^ithprime(k)/ithprime(k)),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 24; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] (-d)^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[PrimeQ[d]] (-d)^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} ( Sum_{p|k, p prime} (-p)^(1-k/p) ) * x^k/k).