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%I #18 Jan 28 2022 15:44:44
%S 1,1,2,2,3,2,5,4,8,8,12,12,19,18,24,26,36,36,48,50,70,71,92,96,129,
%T 133,168,177,225,233,294,307,382,401,488,518,635,668,803,855,1027,
%U 1089,1298,1381,1638,1745,2047,2184,2569,2734,3181,3404,3953,4213,4863,5203
%N Number of partitions of n in which any two distinct parts differ by at least 4.
%C Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 4 times.
%H Vaclav Kotesovec, <a href="/A218699/b218699.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^(4*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(-4*x)) dx = -0.9030055506558938921393786530232872470622617736... - _Vaclav Kotesovec_, Jan 28 2022
%e a(5) = 2: [1,1,1,1,1], [5].
%e a(6) = 5: [1,1,1,1,1,1], [2,2,2], [3,3], [1,5], [6].
%e a(7) = 4: [1,1,1,1,1,1,1], [1,1,5], [1,6], [7].
%e a(8) = 8: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [1,1,1,5], [1,1,6], [2,6], [1,7], [8].
%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-4), j=1..n/i)))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..70);
%t 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 a[n_] := b[n, n, 4];
%t a /@ Range[0, 70] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y Column k=4 of A218698.
%Y Cf. A160974.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Nov 04 2012
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