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A264396 Number of partitions of n such that the part sizes occurring in it form an interval that does not start at 1. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Dec 22 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A251729.

Sequence in context: A084630 A325393 A320388 * A007360 A029144 A173244

Adjacent sequences:  A264393 A264394 A264395 * A264397 A264398 A264399

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 17 2015

STATUS

approved

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Last modified August 19 20:20 EDT 2019. Contains 326133 sequences. (Running on oeis4.)