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A116458 Number of partitions of n into parts congruent to 1, 9, or 11 (mod 14). 0

%I #6 Mar 07 2016 04:36:58

%S 1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,5,5,6,6,7,8,9,10,11,12,12,14,15,

%T 16,17,19,21,22,24,26,29,31,34,36,38,41,44,48,51,55,60,64,68,73,79,84,

%U 91,97,103,110,117,125,133,142,152,163,172,183,196,208,222,236,250,265,281

%N Number of partitions of n into parts congruent to 1, 9, or 11 (mod 14).

%C Also number of partitions of n into distinct parts congruent to 1,2, or 4 (mod 7). Example: a(15)=4 because we have [15],[11,4],[9,4,2] and [8,4,2,1].

%D G. E. Andrews, Number Theory, Dover Publications, 1994 (p. 166, Exercise 7).

%F G.f.=1/product((1-x^(1+14j))(1-x^(9+14j))(1-x^(11+14j)),j=0..infinity). G.f.=product((1+x^(1+7j))(1+x^(2+7j))(1+x^(4+7j)),j=0..infinity).

%F a(n) ~ exp(sqrt(n/7)*Pi) / (2*7^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Mar 07 2016

%e a(15)=4 because we have [15],[11,1,1,1,1],[9,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].

%p g:=product((1+x^(1+7*j))*(1+x^(2+7*j))*(1+x^(4+7*j)),j=0..15): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..77);

%t nmax = 100; CoefficientList[Series[Product[(1+x^(7*k-6))*(1+x^(7*k-5))*(1+x^(7*k-3)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 07 2016 *)

%K nonn

%O 0,10

%A _Emeric Deutsch_, Feb 16 2006

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