The On-Line Encyclopedia of Integer Sequences (OEIS)

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A318694

Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)).
(history; published version)

#10 by Alois P. Heinz at Mon Jul 27 15:51:44 EDT 2020

STATUS

proposed

approved

#9 by Robert Israel at Mon Jul 27 11:18:07 EDT 2020

STATUS

editing

proposed

#8 by Robert Israel at Mon Jul 27 11:18:01 EDT 2020

LINKS

Robert Israel, <a href="/A318694/b318694.txt">Table of n, a(n) for n = 0..447</a>

STATUS

approved

editing

#7 by Bruno Berselli at Wed Jan 09 09:16:14 EST 2019

STATUS

editing

approved

#6 by Paolo P. Lava at Wed Jan 09 06:52:35 EST 2019

MAPLE

seq(n!*coeff(series(mul((1+x^k/k)^tau(k), k=1..100), x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 09 2019

STATUS

approved

editing

#5 by Vaclav Kotesovec at Sat Sep 01 03:53:01 EDT 2018

STATUS

reviewed

approved

#4 by Joerg Arndt at Sat Sep 01 02:07:23 EDT 2018

STATUS

proposed

reviewed

#3 by Ilya Gutkovskiy at Fri Aug 31 18:10:19 EDT 2018

STATUS

editing

proposed

#2 by Ilya Gutkovskiy at Fri Aug 31 17:33:50 EDT 2018

NAME

allocated for Ilya Gutkovskiy

Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)).

DATA

1, 1, 2, 10, 40, 248, 1868, 14516, 131920, 1409040, 15697872, 191687472, 2663239104, 37878672960, 582357866400, 9898540886880, 172534018584960, 3192686545714560, 63844374067107840, 1309775114921541120, 28512040933544970240, 656888836504576112640, 15495311684125737031680

OFFSET

0,3

FORMULA

E.g.f.: Product_{k>=1} (1 + x^k/k)^tau(k), where tau = number of divisors (A000005).

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

MATHEMATICA

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j)/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

nmax = 22; CoefficientList[Series[Product[(1 + x^k/k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, 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, Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

CROSSREFS

Cf. A000005, A007838, A107742, A181541, A318416, A318693.

KEYWORD

allocated

nonn

AUTHOR

Ilya Gutkovskiy, Aug 31 2018

STATUS

approved

editing

#1 by Ilya Gutkovskiy at Fri Aug 31 17:33:50 EDT 2018

NAME

allocated for Ilya Gutkovskiy

KEYWORD

allocated

STATUS

approved