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

1,4

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.

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.

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

Example

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

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<i, 0, add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= 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 *)

Crossrefs

Cf. A251729.

Sequence in context: A084630 A325393 A320388 * A007360 A359320 A029144

Adjacent sequences: A264393 A264394 A264395 * A264397 A264398 A264399

Keyword

nonn

Author

Emeric Deutsch, Nov 17 2015

Status

approved