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A118083 Number of partitions of n such that largest part k occurs at least floor(k/2) times. 2

%I #8 Mar 06 2020 07:22:43

%S 1,1,2,3,4,5,7,8,11,13,17,20,26,30,38,45,55,64,79,91,110,128,152,176,

%T 209,240,282,325,379,434,505,576,666,760,873,993,1139,1290,1473,1668,

%U 1897,2141,2430,2736,3095,3481,3925,4404,4958,5550,6232,6968,7805,8710

%N Number of partitions of n such that largest part k occurs at least floor(k/2) times.

%C Also number of partitions of n such that if the number of parts is k, then the smallest part is at least floor(k/2). Example: a(8)=11 because we have [8],[7,1],[6,2],[5,3],[4,4],[6,1,1],[5,2,1],[4,3,1],[4,2,2],[3,3,2] and [2,2,2,2].

%C Also number of partitions of 2*n into distinct parts with either all parts odd or all parts even. - _Vladeta Jovovic_, Jul 03 2007

%F G.f.=sum(x^(k*floor(k/2))/product(1-x^j, j=1..k), k=1..infinity).

%F a(n) = A000700(2*n) + A000009(n), n>0. - _Vladeta Jovovic_, Jul 03 2007

%F a(n) ~ (2 + sqrt(2)) * exp(sqrt(n/3)*Pi) / (8*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Mar 06 2020

%e a(8)=11 because we have [4,4],[3,3,2],[3,3,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1],[2,2,2,2],[2,2,2,1,1],[2,2,1,1,1,1],[2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1].

%p g:=sum(x^(k*floor(k/2))/product(1-x^j,j=1..k),k=1..15): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60);

%Y Cf. A118082, A118084.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Apr 12 2006

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