A218705 - OEIS

%I #12 Jan 29 2022 04:15:56

%S 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 51,48,61,57,69,65,80,81,98,93,115,112,144,136,170,164,202,204,244,

%U 242,296,290,353,350,415,412,493,494,576,580,671,673,794,786,903

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

%C Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 10 times.

%H 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 G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(10*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(-10*x)) dx = -1.2055372531240537414216314471404302128615809819... - _Vaclav Kotesovec_, Jan 28 2022

%e a(10) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].

%e a(11) = 2: [1,1,1,1,1,1,1,1,1,1,1], [11].

%e 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 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-10), j=1..n/i)))

%p     end:

%p a:= n-> b(n, n):

%p seq(a(n), n=0..70);

%Y Column k=10 of A218698.

%Y Cf. A160980.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 04 2012