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A118199 Number of partitions of n having no parts equal to the size of their Durfee squares. 2

%I

%S 1,0,1,1,1,1,2,3,5,7,10,13,18,23,31,40,53,68,89,113,146,184,234,293,

%T 369,458,572,706,874,1073,1320,1611,1970,2393,2909,3518,4255,5122,

%U 6167,7394,8862,10585,12637,15038,17886,21213,25141,29723,35112,41383,48737,57278

%N Number of partitions of n having no parts equal to the size of their Durfee squares.

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

%H Alois P. Heinz, <a href="/A118199/b118199.txt">Table of n, a(n) for n = 0..1000</a>

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

%e 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.

%p 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);

%p # second Maple program::

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

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

%p end:

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

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

%p seq(a(n), n=0..70); # _Alois P. Heinz_, Apr 09 2012

%t 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_ *)

%Y Cf. A118198, A000041.

%K nonn

%O 0,7

%A _Emeric Deutsch_, Apr 14 2006

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Last modified April 24 00:02 EDT 2019. Contains 322404 sequences. (Running on oeis4.)