A222709 Total number of parts of multiplicity 9 in all partitions of n.

1, 0, 1, 1, 2, 2, 4, 4, 7, 9, 13, 15, 23, 27, 38, 47, 63, 77, 104, 126, 165, 202, 259, 316, 403, 489, 614, 748, 929, 1125, 1391, 1676, 2055, 2475, 3012, 3613, 4379, 5233, 6306, 7521, 9018, 10717, 12805, 15171, 18050, 21337, 25288, 29806, 35221, 41400, 48760

Offset

9,5

Links

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Formula

G.f.: (x^9/(1-x^9)-x^10/(1-x^10))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (180*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=9, 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=9..60);

Mathematica

b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 9, 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, 9, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Crossrefs

Column k=9 of A197126.

Sequence in context: A206559 A143419 A183566 * A034396 A364193 A253412

Adjacent sequences: A222706 A222707 A222708 * A222710 A222711 A222712

Keyword

nonn

Author

Alois P. Heinz, Feb 28 2013

Status

approved