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Number of partitions of n with up to seven distinct kinds of 1.
2

%I #10 Apr 24 2021 20:44:25

%S 1,7,22,43,65,93,137,201,287,401,552,751,1010,1344,1774,2323,3017,

%T 3893,4995,6370,8078,10195,12809,16023,19958,24761,30613,37720,46317,

%U 56693,69192,84207,102200,123715,149384,179947,216265,259338,310333,370593,441667

%N Number of partitions of n with up to seven distinct kinds of 1.

%H Alois P. Heinz, <a href="/A320694/b320694.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ Pi * 2^(9/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018

%F G.f.: (1 + x)^7 * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021

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

%p binomial(7, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))

%p end:

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

%p seq(a(n), n=0..60);

%Y Column k=7 of A292622.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 19 2018