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
Keywords
Examples
a(2) = 4: 2, 1a1a, 1a1b, 1b1b.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1664
Programs
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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 *)
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