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A118096
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Number of partitions of n such that the largest part is twice the smallest part.
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32
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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, 48, 49, 61, 57, 68, 72, 78, 81, 97, 95, 107, 114, 127, 128, 150, 148, 168, 179, 191, 198, 229, 230, 254, 266, 291, 300, 338, 344, 379, 398, 427, 444, 498, 505, 550, 580, 625
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OFFSET
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1,5
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COMMENTS
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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].
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(3*k)/Product_{j=k..2*k} (1-x^j).
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EXAMPLE
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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].
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MAPLE
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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);
# second Maple program:
b:= proc(n, i, t) option remember: `if`(n=0, 1, `if`(i<t, 0,
b(n, i-1, t)+`if`(i>n, 0, b(n-i, i, t))))
end:
a:= n-> add(b(n-3*j, 2*j, j), j=1..n/3):
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MATHEMATICA
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Table[Count[IntegerPartitions[n], p_ /; 2 Min[p] = = Max[p]], {n, 40}] (* Clark Kimberling, Feb 16 2014 *)
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < t, 0,
b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t]]]];
a[n_] := Sum[b[n - 3j, 2j, j], {j, 1, n/3}];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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