The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A298906 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A298906

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

#14 by Joerg Arndt at Thu Jul 01 11:48:34 EDT 2021

STATUS

proposed

approved

#13 by Seiichi Manyama at Thu Jul 01 11:39:50 EDT 2021

STATUS

editing

proposed

#12 by Seiichi Manyama at Thu Jul 01 11:36:20 EDT 2021

LINKS

Seiichi Manyama, <a href="/A298906/b298906.txt">Table of n, a(n) for n = 0..450</a>

#11 by Seiichi Manyama at Thu Jul 01 11:24:52 EDT 2021

CROSSREFS

Cf. A000110, A168243, A209902, A345762.

STATUS

approved

editing

#10 by Bruno Berselli at Wed Mar 27 03:51:24 EDT 2019

STATUS

editing

approved

#9 by Paolo P. Lava at Tue Mar 26 06:58:33 EDT 2019

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

STATUS

approved

editing

#8 by Susanna Cuyler at Tue Jun 19 22:12:44 EDT 2018

STATUS

proposed

approved

#7 by Ilya Gutkovskiy at Mon Jun 18 15:18:56 EDT 2018

STATUS

editing

proposed

#6 by Ilya Gutkovskiy at Mon Jun 18 14:17:41 EDT 2018

FORMULA

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).

#5 by Ilya Gutkovskiy at Mon Jun 18 14:16:33 EDT 2018

NAME

allocated for Ilya Gutkovskiy

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

DATA

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

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!) * ...

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.

KEYWORD

allocated

sign

AUTHOR

Ilya Gutkovskiy, Jun 18 2018

STATUS

approved

editing