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A218705 Number of partitions of n in which any two distinct parts differ by at least 10.

%I A218705 #12 Jan 29 2022 04:15:56
%S A218705 1,1,2,2,3,2,4,2,4,3,4,2,7,4,8,9,13,11,18,16,23,22,27,27,36,35,41,42,
%T A218705 51,48,61,57,69,65,80,81,98,93,115,112,144,136,170,164,202,204,244,
%U A218705 242,296,290,353,350,415,412,493,494,576,580,671,673,794,786,903
%N A218705 Number of partitions of n in which any two distinct parts differ by at least 10.
%C A218705 Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 10 times.
%H A218705 Vaclav Kotesovec, <a href="/A218705/b218705.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A218705 G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(10*i)/(1-x^i)).
%F A218705 log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-10*x)) dx = -1.2055372531240537414216314471404302128615809819... - _Vaclav Kotesovec_, Jan 28 2022
%e A218705 a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
%e A218705 a(11) = 2: [1,1,1,1,1,1,1,1,1,1,1], [11].
%e A218705 a(12) = 7: [1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [1,11], [12].
%p A218705 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A218705        b(n, i-1) +add(b(n-i*j, i-10), j=1..n/i)))
%p A218705     end:
%p A218705 a:= n-> b(n, n):
%p A218705 seq(a(n), n=0..70);
%Y A218705 Column k=10 of A218698.
%Y A218705 Cf. A160980.
%K A218705 nonn
%O A218705 0,3
%A A218705 _Alois P. Heinz_, Nov 04 2012