A292463 Number of partitions of n with n kinds of 1.

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

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

Equivalently, c = 1/(4*sqrt(Pi)*QPochhammer(1/2)). - Vaclav Kotesovec, Mar 17 2024

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 * A371870 A149488 A058692

Adjacent sequences: A292460 A292461 A292462 * A292464 A292465 A292466

Keyword

nonn

Author

Alois P. Heinz, Sep 16 2017

Status

approved