A264396 Number of partitions of n such that the part sizes occurring in it form an interval that does not start at 1.

0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 9, 12, 12, 14, 17, 18, 21, 25, 26, 30, 36, 39, 43, 51, 55, 62, 73, 78, 88, 101, 110, 125, 141, 154, 172, 195, 215, 238, 269, 294, 327, 368, 402, 446, 498, 547, 606, 672, 737, 814, 903, 991, 1091, 1205, 1320, 1452, 1603, 1752, 1924, 2118, 2314, 2539, 2785, 3042, 3329, 3648, 3984

Offset: 1

Author

Emeric Deutsch, Nov 17 2015

Keywords

• nonn

Comments

The partitions in the definition are called non-complete gap-free (see the Grabner et al. reference).

a(n) = number of partitions of n where the largest part occurs at least twice and all other parts are distinct. Example: a(9) = 5 because we have 441, 333, 3321, 22221, and 111111111.

Examples

			a(9) = 5 because there are these partitions of 9: 9, 54, 432, 333, and 3222.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

Crossrefs

Cf. A251729.

Programs

Maple

g := sum(x^(2*j)*(product(1+x^i, i = 1 .. j-1))/(1-x^j), j = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 1 .. 70);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, 0, 1),
      `if`(i<1 or n add(b(n, i), i=2..n):
seq(a(n), n=1..70);  # Alois P. Heinz, Nov 29 2015

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, If[i == 0, 0, 1], If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; a[n_] := Sum[b[n, i], {i, 2, n}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

G.f.: g = sum((x^{2j}/(1-x^j))*product(1+x^i, i=1..j), j=1..infinity).

a(n) ~ 3^(1/4) * Pi * exp(Pi*sqrt(n/3)) / (24 * n^(5/4)). - Vaclav Kotesovec, May 24 2018