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%I #23 Sep 18 2020 08:47:06
%S 1,0,0,0,1,1,1,1,2,1,2,1,4,2,4,4,7,5,8,7,13,10,13,12,21,18,22,21,34,
%T 29,40,36,55,48,63,64,88,79,100,99,139,125,160,155,207,199,241,241,
%U 314,302,369,366,466,454,550,557,690,679,807,821,1016,1001,1180,1207,1460,1466,1708
%N Number of partitions of n where every part appears at least 4 times.
%H Alois P. Heinz, <a href="/A160974/b160974.txt">Table of n, a(n) for n = 0..10000</a> (terms n=1..967 from R. H. Hardin)
%F G.f.: Product_{j>=1} (1+x^(4*j)/(1-x^j)). - _Emeric Deutsch_, Jun 24 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(-4*x)) dx = -0.903005550655893892139378653023287247062261773608753265529... . - _Vaclav Kotesovec_, Jan 05 2016
%e a(12) = 4 because we have 3333, 2^6, 22221111, and 1^(12). - _Emeric Deutsch_, Jun 24 2009
%p g := product(1+x^(4*j)/(1-x^j), j = 1..30): gser := series(g, x = 0, 85): seq(coeff(gser, x, n), n = 0..66); # _Emeric Deutsch_, Jun 24 2009
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1), j=[0, $4..iquo(n, i)])))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Oct 02 2017
%t nmax = 100; CoefficientList[Series[Product[1 + x^(4*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 28 2015; offset adapted by _Georg Fischer_, Sep 18 2020 *)
%Y Cf. A007690, A100405, A160975-A160990.
%K nonn
%O 0,9
%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 _Alois P. Heinz_, Oct 02 2017
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