A218703 - OEIS

%I #12 Jan 29 2022 04:14:57

%S 1,1,2,2,3,2,4,2,4,3,5,4,10,7,12,13,17,16,23,21,30,30,34,35,47,43,51,

%T 52,66,63,81,77,100,99,120,121,156,150,185,189,234,230,283,281,343,

%U 350,409,414,503,497,587,600,695,703,824,830,967,988,1122,1148,1333

%N Number of partitions of n in which any two distinct parts differ by at least 8.

%C Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 8 times.

%H Vaclav Kotesovec, <a href="/A218703/b218703.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^(8*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(-8*x)) dx = -1.1447921975208768146551512630331558734964408879... - _Vaclav Kotesovec_, Jan 28 2022

%e a(9) = 3: [1,1,1,1,1,1,1,1,1], [3,3,3], [9].

%e a(10) = 5: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,9], [10].

%e a(11) = 4: [1,1,1,1,1,1,1,1,1,1,1], [1,1,9], [1,10], [11].

%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-8), j=1..n/i)))

%p     end:

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

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

%Y Column k=8 of A218698.

%Y Cf. A160978.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 04 2012