A222704 Total number of parts of multiplicity 4 in all partitions of n.

1, 0, 1, 1, 3, 3, 5, 6, 11, 13, 20, 24, 37, 45, 64, 80, 110, 137, 184, 229, 303, 375, 486, 602, 772, 951, 1202, 1478, 1853, 2267, 2817, 3432, 4236, 5142, 6300, 7620, 9284, 11185, 13553, 16273, 19625, 23478, 28187, 33613, 40192, 47778, 56904, 67443, 80051

Offset

4,5

Links

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Formula

G.f.: (x^4/(1-x^4)-x^5/(1-x^5))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (40*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

Maple

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=4, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=4..60);

Mathematica

b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m == 4, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 4, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Crossrefs

Column k=4 of A197126.

Sequence in context: A276434 A183561 A300183 * A095950 A089874 A092035

Adjacent sequences: A222701 A222702 A222703 * A222705 A222706 A222707

Keyword

nonn

Author

Alois P. Heinz, Feb 28 2013

Status

approved