A319113 - OEIS

%I #10 Feb 27 2022 12:01:58

%S 1,0,1,2,0,44,0,1224,2688,25920,293760,3628800,25090560,762048000,

%T 3887170560,62749209600,1233908121600,22616539545600,321930878976000,

%U 10717413809356800,108951843667968000,1982497256570880000,50138292140310528000,1408088823809310720000,25175914255793258496000

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

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

%p 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

%t nmax = 24; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t 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]!

%t 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}]

%o (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

%Y Cf. A002099, A007838, A319112.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Sep 10 2018