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A218704 Number of partitions of n in which any two distinct parts differ by at least 9.

%I A218704 #13 Jan 29 2022 04:15:27
%S A218704 1,1,2,2,3,2,4,2,4,3,4,3,8,6,9,12,14,14,20,19,24,28,29,31,41,40,44,49,
%T A218704 54,54,69,64,77,82,94,98,119,118,139,149,173,178,215,217,253,274,306,
%U A218704 320,375,385,440,469,521,545,626,647,718,769,838,881,994,1026
%N A218704 Number of partitions of n in which any two distinct parts differ by at least 9.
%C A218704 Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 9 times.
%H A218704 Vaclav Kotesovec, <a href="/A218704/b218704.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A218704 G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(9*i)/(1-x^i)).
%F A218704 log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-9*x)) dx = -1.1777463281753839250026821894706693162991897931... - _Vaclav Kotesovec_, Jan 28 2022
%e A218704 a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
%e A218704 a(11) = 3: [1,1,1,1,1,1,1,1,1,1,1], [1,10], [11].
%p A218704 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A218704        b(n, i-1) +add(b(n-i*j, i-9), j=1..n/i)))
%p A218704     end:
%p A218704 a:= n-> b(n, n):
%p A218704 seq(a(n), n=0..70);
%Y A218704 Column k=9 of A218698.
%Y A218704 Cf. A160979.
%K A218704 nonn
%O A218704 0,3
%A A218704 _Alois P. Heinz_, Nov 04 2012