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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.
1
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
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
Sequence in context: A036709 A052115 A007882 * A009852 A323951 A080003
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jan 11 2016
STATUS
approved