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A118096
Number of partitions of n such that the largest part is twice the smallest part.
32
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
OFFSET
1,5
COMMENTS
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].
LINKS
FORMULA
G.f.: Sum_{k>=1} x^(3*k)/Product_{j=k..2*k} (1-x^j).
EXAMPLE
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].
MAPLE
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):
seq(a(n), n=1..64); # Alois P. Heinz, Sep 04 2017
MATHEMATICA
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}];
Array[a, 64] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 12 2006
STATUS
approved