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Number of partitions of n such that the largest part is twice the smallest part.
32

%I #31 May 14 2023 23:47:50

%S 0,0,1,1,2,3,3,4,6,6,6,10,9,11,13,14,15,20,18,23,25,27,27,37,35,39,43,

%T 48,49,61,57,68,72,78,81,97,95,107,114,127,128,150,148,168,179,191,

%U 198,229,230,254,266,291,300,338,344,379,398,427,444,498,505,550,580,625

%N Number of partitions of n such that the largest part is twice the smallest part.

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

%H Alois P. Heinz, <a href="/A118096/b118096.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{k>=1} x^(3*k)/Product_{j=k..2*k} (1-x^j).

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

%p g:=sum(x^(3*k)/product(1-x^j,j=k..2*k),k=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..70);

%p # second Maple program:

%p b:= proc(n, i, t) option remember: `if`(n=0, 1, `if`(i<t, 0,

%p b(n, i-1, t)+`if`(i>n, 0, b(n-i, i, t))))

%p end:

%p a:= n-> add(b(n-3*j, 2*j, j), j=1..n/3):

%p seq(a(n), n=1..64); # _Alois P. Heinz_, Sep 04 2017

%t Table[Count[IntegerPartitions[n], p_ /; 2 Min[p] = = Max[p]], {n, 40}] (* _Clark Kimberling_, Feb 16 2014 *)

%t (* Second program: *)

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < t, 0,

%t b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t]]]];

%t a[n_] := Sum[b[n - 3j, 2j, j], {j, 1, n/3}];

%t Array[a, 64] (* _Jean-François Alcover_, Jun 04 2021, after _Alois P. Heinz_ *)

%o (PARI) my(N=70, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=k, 2*k, 1-x^j)))) \\ _Seiichi Manyama_, May 14 2023

%Y Cf. A237825, A237826, A237827.

%Y Cf. A008483, A117086, A238479, A350893.

%K nonn

%O 1,5

%A _Emeric Deutsch_, Apr 12 2006