|
%I #13 Feb 06 2024 18:46:33
%S 1,0,0,0,0,0,0,1,1,1,1,1,1,1,2,1,2,1,2,1,2,3,3,3,5,4,5,6,8,7,9,9,11,
%T 11,12,15,16,16,17,19,22,21,26,26,30,31,34,36,43,46,50,52,60,62,70,74,
%U 89,88,99,104,118,122,136,150,163,169,187,196,216,227,256,264,295,304,332,350,382,411,441
%N Number of partitions of n where every part appears at least 7 times.
%H Alois P. Heinz, <a href="/A160977/b160977.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..1000 from R. H. Hardin)
%F a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-7*x)) dx = -1.104868234083422137620242346741601264555358762045898765433... . - _Vaclav Kotesovec_, Jan 05 2016
%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-1), j=7..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..78); # _Alois P. Heinz_, Feb 06 2024
%t nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(7*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%K nonn
%O 0,15
%A _R. H. Hardin_, Jun 01 2009
%E a(0)=1 prepended by _Alois P. Heinz_, Feb 06 2024
|
