The On-Line Encyclopedia of Integer Sequences (OEIS)

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A317912

Expansion of Product_{k>=2} 1/(1 - k*x^k).
(history; published version)

#10 by Vaclav Kotesovec at Tue Dec 08 11:40:33 EST 2020

STATUS

reviewed

approved

#9 by Michel Marcus at Tue Dec 08 10:49:43 EST 2020

STATUS

proposed

reviewed

#8 by Seiichi Manyama at Tue Dec 08 10:41:09 EST 2020

STATUS

editing

proposed

#7 by Seiichi Manyama at Tue Dec 08 10:38:43 EST 2020

LINKS

Seiichi Manyama, <a href="/A317912/b317912.txt">Table of n, a(n) for n = 0..5000</a>

STATUS

approved

editing

#6 by Bruno Berselli at Fri Aug 10 17:32:16 EDT 2018

STATUS

proposed

approved

#5 by Alois P. Heinz at Fri Aug 10 15:58:18 EDT 2018

STATUS

editing

proposed

#4 by Alois P. Heinz at Fri Aug 10 15:58:15 EDT 2018

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

b(n, i-1)+ i*b(n-i, min(n-i, i)))

end:

a:= n-> b(n$2) -`if`(n=0, 0, b(n-1$2)):

seq(a(n), n=0..40); # Alois P. Heinz, Aug 10 2018

STATUS

proposed

editing

#3 by Ilya Gutkovskiy at Fri Aug 10 15:42:41 EDT 2018

STATUS

editing

proposed

#2 by Ilya Gutkovskiy at Fri Aug 10 15:10:13 EDT 2018

NAME

allocated for Ilya Gutkovskiy

Expansion of Product_{k>=2} 1/(1 - k*x^k).

DATA

1, 0, 2, 3, 8, 11, 31, 41, 101, 156, 318, 498, 1037, 1555, 3024, 4889, 8849, 14112, 25622, 40322, 71314, 113926, 194677, 310819, 530030, 835484, 1400523, 2226307, 3668998, 5797558, 9521310, 14942262, 24298136, 38187102, 61384028, 96161997, 154078991, 239891926, 381723396

OFFSET

0,3

COMMENTS

First differences of A006906.

Sum of products of terms in all partitions of n into parts >= 2.

LINKS

<a href="/index/Par#part">Index entries for sequences related to partitions</a>

FORMULA

G.f.: exp(Sum_{j>=1} Sum_{k>=2} k^j*x^(j*k)/j).

EXAMPLE

a(6) = 31 because we have [6], [4, 2], [3, 3], [2, 2, 2] and 6 + 4*2 + 3*3 + 2*2*2 = 31.

MATHEMATICA

nmax = 38; CoefficientList[Series[Product[1/(1 - k x^k), {k, 2, nmax}], {x, 0, nmax}], x]

nmax = 38; CoefficientList[Series[Exp[Sum[Sum[k^j x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 38}]]

CROSSREFS

Cf. A002865, A006906, A191659, A302830.

KEYWORD

allocated

nonn

AUTHOR

Ilya Gutkovskiy, Aug 10 2018

STATUS

approved

editing

#1 by Ilya Gutkovskiy at Fri Aug 10 15:10:13 EDT 2018

NAME

allocated for Ilya Gutkovskiy

KEYWORD

allocated

STATUS

approved