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A218700 Number of partitions of n in which any two distinct parts differ by at least 5.

%I A218700 #16 Jan 28 2022 15:46:04
%S A218700 1,1,2,2,3,2,4,3,6,7,9,10,15,15,19,23,26,28,36,37,48,52,62,67,85,93,
%T A218700 110,122,144,157,194,205,241,265,304,338,391,422,483,533,607,661,760,
%U A218700 822,933,1032,1151,1260,1432,1554,1751,1920,2137,2333,2621,2848,3176
%N A218700 Number of partitions of n in which any two distinct parts differ by at least 5.
%C A218700 Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 5 times.
%H A218700 Vaclav Kotesovec, <a href="/A218700/b218700.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A218700 G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(5*i)/(1-x^i)).
%F A218700 log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-5*x)) dx = -0.9908078441778424729564846063206238729218368028... - _Vaclav Kotesovec_, Jan 28 2022
%e A218700 a(6) = 4: [1,1,1,1,1,1], [2,2,2], [3,3], [6].
%e A218700 a(7) = 3: [1,1,1,1,1,1,1], [1,6], [7].
%e A218700 a(8) = 6: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,6], [1,7], [8].
%p A218700 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A218700        b(n, i-1) +add(b(n-i*j, i-5), j=1..n/i)))
%p A218700     end:
%p A218700 a:= n-> b(n, n):
%p A218700 seq(a(n), n=0..70);
%t A218700 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i j, i - k, k], {j, 1, n/i}]]];
%t A218700 a[n_] := b[n, n, 5];
%t A218700 a /@ Range[0, 70] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y A218700 Column k=5 of A218698.
%Y A218700 Cf. A160975.
%K A218700 nonn
%O A218700 0,3
%A A218700 _Alois P. Heinz_, Nov 04 2012