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A118199 Number of partitions of n having no parts equal to the size of their Durfee squares. 2
1, 0, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 40, 53, 68, 89, 113, 146, 184, 234, 293, 369, 458, 572, 706, 874, 1073, 1320, 1611, 1970, 2393, 2909, 3518, 4255, 5122, 6167, 7394, 8862, 10585, 12637, 15038, 17886, 21213, 25141, 29723, 35112, 41383, 48737, 57278 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

a(n) = A118198(n,0).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).

EXAMPLE

a(7) = 3 because we have [7] with size of Durfee square 1, [4,3] with size of Durfee square 2 and [3,3,1] with size of Durfee square 2.

MAPLE

g:=1+sum(x^(k^2+k)/(1-x^k)/product((1-x^i)^2, i=1..k-1), k=1..20): gser:=series(g, x=0, 60): seq(coeff(gser, x, n), n=0..54);

# second Maple program::

b:= proc(n, i) option remember; `if`(n=0, 1,

      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

    end:

a:= n-> add(add(b(k, d) *b(n-d*(d+1)-k, d-1),

                k=0..n-d*(d+1)), d=0..floor(sqrt(n))):

seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012

MATHEMATICA

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[Sum[b[k, d]*b[n-d*(d+1)-k, d-1], {k, 0, n-d*(d+1)}], {d, 0, Floor[Sqrt[n]]}]; Table[a[n], {n, 0, 70}] (* Jean-Fran├žois Alcover, May 22 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A118198, A000041.

Sequence in context: A001401 A008628 A038499 * A239883 A088318 A038083

Adjacent sequences:  A118196 A118197 A118198 * A118200 A118201 A118202

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 14 2006

STATUS

approved

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Last modified March 19 15:02 EDT 2019. Contains 321330 sequences. (Running on oeis4.)