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%I #22 Feb 06 2024 09:56:24
%S 1,0,0,0,0,1,1,1,1,1,2,1,2,1,2,3,3,3,5,4,7,7,7,8,11,12,12,14,15,16,23,
%T 20,24,26,29,36,40,40,46,50,63,63,76,76,87,103,108,117,135,140,167,
%U 173,191,205,235,257,278,300,327,354,413,424,469,511,555,616,673,711,783,849,947
%N Number of partitions of n where every part appears at least 5 times.
%H Seiichi Manyama, <a href="/A160975/b160975.txt">Table of n, a(n) for n = 0..5000</a> (terms 1..1000 from R. H. Hardin)
%F G.f.: Product_{j>=1} (1+x^(5*j)/(1-x^j)). - _Emeric Deutsch_, Jun 28 2009
%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(-5*x)) dx = -0.990807844177842472956484606320623872921836802804155824925... . - _Vaclav Kotesovec_, Jan 05 2016
%e a(15) = 3 because we have 33333, 2222211111, and 1^(15). - _Emeric Deutsch_, Jun 28 2009
%p g := product(1+x^(5*j)/(1-x^j), j = 1..20): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0..75); # _Emeric Deutsch_, Jun 28 2009
%p # second Maple program:
%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=5..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..70); # _Alois P. Heinz_, Feb 06 2024
%t nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(5*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%Y Cf. A007690, A100405, A160974, A160976-A160990.
%K nonn
%O 0,11
%A _R. H. Hardin_, Jun 01 2009
%E Initial terms changed to match b-file. - _N. J. A. Sloane_, Aug 31 2009
%E Maple program fixed by _Vaclav Kotesovec_, Nov 28 2015
%E a(0)=1 prepended by _Seiichi Manyama_, Feb 06 2024
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