A298906 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(1/k!).

1, 1, 1, 4, 2, 1, 77, 29, -4289, -14836, 283812, 1316855, -16548717, -292820579, 911200565, 52594983250, 100157634380, -3444629077653, 7961210574683, -2170805244559295, -41176659971108705, 348776485253486302, 35663019455311634058, 513993485453689440281

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..450

Formula

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

E.g.f.: Product_{k>=1} B(x^k)^((-1)^(k+1)/k), where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers (A000110).

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 2*x^4/4! + x^5/5! + 77*x^6/6! + ... = (1 + x) * (1 + x^2)^(1/2!) * (1 + x^3)^(1/3!) * (1 + x^4)^(1/4!) * ...

Maple

a:=series(exp(add((-1)^(k+1)*(exp(x^k)-1)/k, k=1..100)), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 23; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (Exp[x^k] - 1)/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1)/(d - 1)!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 23}]

Crossrefs

Cf. A000110, A168243, A209902, A345762.

Sequence in context: A136215 A136737 A275595 * A004551 A284398 A285001

Adjacent sequences: A298903 A298904 A298905 * A298907 A298908 A298909

Keyword

sign

Author

Ilya Gutkovskiy, Jun 18 2018

Status

approved