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A238590
Number of partitions p of n such that 3*min(p) is a part of p.
4
0, 0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 32, 46, 61, 86, 110, 149, 192, 257, 326, 425, 538, 694, 871, 1107, 1381, 1740, 2154, 2689, 3313, 4103, 5024, 6176, 7529, 9201, 11157, 13554, 16365, 19784, 23782, 28610, 34260, 41039, 48958, 58405, 69431, 82525, 97775
OFFSET
1,6
LINKS
FORMULA
G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023
EXAMPLE
a(7) = 3 counts these partitions: 331, 3211, 31111.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, b(n, i+1)+b(n-i, i)))
end:
a:= n-> add(b(n-4*i, i), i=1..n/4):
seq(a(n), n=1..60); # Alois P. Heinz, Mar 03 2014
MATHEMATICA
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Min[p]]], {n, 50}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n-4i, i], {i, 1, n/4}];
Array[a, 60] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
PROG
(PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 01 2014
STATUS
approved