A265258 Total number of corners in all partitions of n. A corner of a partition is a point of degree two in the corresponding Ferrers diagram.

4, 8, 13, 22, 33, 52, 75, 111, 157, 223, 307, 426, 575, 778, 1036, 1377, 1806, 2367, 3067, 3968, 5090, 6512, 8273, 10488, 13212, 16604, 20762, 25896, 32155, 39837, 49155, 60518, 74249, 90893, 110922, 135087, 164044, 198815, 240340, 289984, 349057, 419413, 502848, 601851, 718903

Offset

1,1

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, Counting corners in partitions, Ramanujan J., 39, 2016, 201-224.

Formula

G.f.: G(x) = -3 + (3 - 2x)/(1-x)/Product_{j>=1} (1 - x^j).

a(n) = 3*p(n) + Sum_{j=1..n} p(n - j), where p(n) = A000041(n) is the number of partitions of n.

a(n) = A000070(n) + 2*A000041(n). - Vaclav Kotesovec, Jan 11 2016

a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(8*n)). - Vaclav Kotesovec, Jan 11 2016

Example

a(1) = 4 because, obviously, the number of corners of the partition [1] is 4.
a(3) = 13 because the number of corners of the partitions [3], [2,1], and [1,1,1] are 4, 5, and 4, respectively.

Maple

with(combinat): p := numbpart: seq(3*p(n)+add(p(n-j), j=1..n), n=1..45);
# 2nd Maple program:
g := -3+(3-2*x)/((1-x)*mul(1-x^k, k = 1 .. 100)): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 1 .. 45);
# third Maple program:
p:= proc(n) option remember; `if`(n=0, 1, add(add(
       d, d=numtheory[divisors](j))*p(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=1, 4,
       a(n-1) +3*p(n) -2*p(n-1))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 11 2016

Mathematica

p = PartitionsP; Table[3*p[n] + Sum[p[n-j], {j, 1, n}], {n, 1, 45}] (* Jean-François Alcover, Jan 24 2016, adapted from 1st Maple program *)

Crossrefs

Cf. A000041, A000070.

Sequence in context: A036709 A052115 A007882 * A009852 A323951 A080003

Adjacent sequences: A265255 A265256 A265257 * A265259 A265260 A265261

Keyword

nonn

Author

Emeric Deutsch, Jan 11 2016

Status

approved