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A292463 Number of partitions of n with n kinds of 1. 9
1, 1, 4, 14, 51, 188, 702, 2644, 10026, 38223, 146359, 562456, 2168134, 8379539, 32459199, 125984039, 489837300, 1907490728, 7438346255, 29042470132, 113522618066, 444199913556, 1739735079466, 6819657196928, 26753893533257, 105034060120469, 412637434996367 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1664

FORMULA

a(n) = [x^n] 1/(1-x)^n * 1/Product_{j=2..n} (1-x^j).

a(n) is n-th term of the Euler transform of n,1,1,1,... .

a(n) ~ c * 4^n / sqrt(n), where c = QPochhammer[-1, 1/2] / (8*sqrt(Pi) * QPochhammer[1/4, 1/4]) = 0.48841139329043831428669851139824427133317... - Vaclav Kotesovec, Sep 19 2017

EXAMPLE

a(2) = 4: 2, 1a1a, 1a1b, 1b1b.

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0 or i<2,

      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))

    end:

a:= n-> b(n$3):

seq(a(n), n=0..30);

# second Maple program:

b:= proc(n, k) option remember; `if`(n=0, 1, add(

      (numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)

    end:

a:= n-> b(n$2):

seq(a(n), n=0..30);

# third Maple program:

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

      combinat[numbpart](n), b(n-1, k) +b(n, k-1)))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..30);

MATHEMATICA

Table[SeriesCoefficient[1/(1-x)^(n-1) * Product[1/(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 19 2017 *)

CROSSREFS

Main diagonal of A292508.

Cf. A000041, A014329, A292462, A292503, A292507, A292541.

Sequence in context: A283108 A211303 A247415 * A149488 A058692 A165813

Adjacent sequences:  A292460 A292461 A292462 * A292464 A292465 A292466

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 16 2017

STATUS

approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)