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A218701 Number of partitions of n in which any two distinct parts differ by at least 6.

%I A218701 #12 Jan 28 2022 15:47:03
%S A218701 1,1,2,2,3,2,4,2,5,5,8,7,14,11,16,19,23,22,32,29,38,40,48,48,67,63,81,
%T A218701 85,106,106,141,138,174,180,219,224,284,282,342,356,422,431,530,532,
%U A218701 631,660,765,789,948,965,1123,1184,1356,1408,1658,1703,1967,2076
%N A218701 Number of partitions of n in which any two distinct parts differ by at least 6.
%C A218701 Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 6 times.
%H A218701 Vaclav Kotesovec, <a href="/A218701/b218701.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A218701 G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(6*i)/(1-x^i)).
%F A218701 log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-6*x)) dx = -1.0551351195231385243969621008395374852115209081... - _Vaclav Kotesovec_, Jan 28 2022
%e A218701 a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
%e A218701 a(7) = 2: [1,1,1,1,1,1,1], [7].
%e A218701 a(8) = 5: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,7], [8].
%p A218701 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A218701        b(n, i-1) +add(b(n-i*j, i-6), j=1..n/i)))
%p A218701     end:
%p A218701 a:= n-> b(n, n):
%p A218701 seq(a(n), n=0..70);
%Y A218701 Column k=6 of A218698.
%Y A218701 Cf. A160976.
%K A218701 nonn
%O A218701 0,3
%A A218701 _Alois P. Heinz_, Nov 04 2012