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%I #12 Jan 29 2022 04:14:23
%S 1,1,2,2,3,2,4,2,4,4,6,6,11,10,13,16,19,20,25,27,33,34,39,41,51,52,61,
%T 65,80,82,99,104,126,133,156,168,199,209,243,261,302,320,372,392,447,
%U 479,537,572,650,693,770,829,920,982,1096,1169,1306,1396,1541
%N Number of partitions of n in which any two distinct parts differ by at least 7.
%C Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 7 times.
%H Vaclav Kotesovec, <a href="/A218702/b218702.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(7*i)/(1-x^i)).
%F log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-7*x)) dx = -1.104868234083422137620242346741601264555358762... - _Vaclav Kotesovec_, Jan 28 2022
%e a(8) = 4: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8].
%e a(9) = 4: [1,1,1,1,1,1,1,1,1], [3,3,3], [1,8], [9].
%e a(10) = 6: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,1,8], [1,9], [10].
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +add(b(n-i*j, i-7), j=1..n/i)))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..70);
%Y Column k=7 of A218698.
%Y Cf. A160977.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Nov 04 2012
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