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A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

%I A025158 #16 Jan 02 2016 12:12:25
%S A025158 1,1,1,1,1,2,2,3,3,4,4,5,5,6,7,8,9,11,12,14,16,18,20,23,25,28,31,35,
%T A025158 38,43,47,53,58,65,71,80,87,97,106,118,128,142,154,170,185,203,220,
%U A025158 242,262,287,311,340,368,402,435,474,513,558,603,656,708,768,829,898,968,1048
%N A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.
%C A025158 Also number of partitions of n such that if k is the largest part, then each 1,2,...,k-1 occur at least 4 times. Example: a(8)=3 because we have [2,2,1,1,1,1], [2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 17 2006
%H A025158 Alois P. Heinz, <a href="/A025158/b025158.txt">Table of n, a(n) for n = 1..1000</a>
%F A025158 G.f.: Sum(x^(2*k^2-k)/Product(1-x^i, i=1..k), k=1..infinity). - _Vladeta Jovovic_, Aug 12 2004
%F A025158 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*sqrt(Pi*r^3*(1+4*r^3))), where r = 0.72449195900051561158837228218703656578649448135001101727... is the root of the equation r^4 + r = 1 and c = 2*log(r)^2 + polylog(2, 1-r) = 0.50498141294472195442598916817438524920370382784609501495065... . - _Vaclav Kotesovec_, Jan 02 2016
%e A025158 a(8) = 3 because we have [8], [7,1] and [6,2].
%p A025158 g:=sum(x^(2*k^2-k)/product(1-x^j,j=1..k),k=1..7): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..66); # _Emeric Deutsch_, Apr 17 2006
%t A025158 nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(j*(2*j - 1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jan 02 2016 *)
%Y A025158 Cf. A003114, A025157-A025162.
%Y A025158 Column k=4 of A194543.
%K A025158 nonn
%O A025158 1,6
%A A025158 _Clark Kimberling_
%E A025158 More terms from _Vladeta Jovovic_, Aug 12 2004