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A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))). 20


%S 1,2,3,4,1,5,2,6,5,7,8,8,14,9,20,1,10,30,2,11,40,5,12,55,10,13,70,18,

%T 14,91,30,15,112,49,16,140,74,1,17,168,110,2,18,204,158,5,19,240,221,

%U 10,20,285,302,20,21,330,407,34,22,385,536,59,23,440,698,94,24,506,896,149,25

%N Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

%C Row n has floor(sqrt(n)) terms. Row sums yield A000041. Column 2 yields A006918. sum(k*T(n,k),k=1..floor(sqrt(n)))=A115995.

%C T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.

%C Successive columns approach closer and closer to A000712. - _N. J. A. Sloane_, Mar 10 2007

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).

%D G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

%D E. R. Canfield, From recursions to asymptotics: Durfee and dilogarithmic deductions, Advances in Applied Mathematics, Volume 34, Issue 4, May 2005, Pages 768-797

%D E. R. Canfield, S. Corteel, C. D. Savage, Electronic Journal of Combinatorics 5 (1998), #R32; http://www.emis.ams.org/journals/EJC/Volume_5/PDF/v5i1r32.pdf

%H Alois P. Heinz, <a href="/A115994/b115994.txt">Rows n = 1..620, flattened</a>

%H S. B. Ekhad, D. Zeilberger, <a href="http://arxiv.org/abs/1411.0002">A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage)</a>, arXiv preprint arXiv:1411.0002, 2014.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, 2009, page 45

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square.</a>

%F G.f.: G(t,q) = sum(t^k*q^(k^2)/product((1-q^j)^2,j=1..k), k=1..infinity).

%F T(n,k) = Sum_{i=0}^{n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.

%e T(5,2) = 2 because the only partitions of 5 having Durfee square of size 2 are [3,2] and [2,2,1]; the other five partitions ([5], [4,1], [3,1,1], [2,1,1,1] and [1,1,1,1,1]) have Durfee square of size 1.

%e Triangle starts:

%e 1;

%e 2;

%e 3;

%e 4, 1;

%e 5, 2;

%e 6, 5;

%e 7, 8;

%e 8, 14;

%e 9, 20, 1;

%e ...

%p g:=sum(t^k*q^(k^2)/product((1-q^j)^2,j=1..k),k=1..40): gser:=series(g,q=0,32): for n from 1 to 27 do P[n]:=coeff(gser,q^n) od: for n from 1 to 27 do seq(coeff(P[n],t^j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form

%p # second Maple program

%p b:= proc(n, i) option remember;

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

%p end:

%p T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):

%p seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # _Alois P. Heinz_, Apr 09 2012

%t Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[ Series[x^(n^2)/Product[1-x^i,{i,1,n}]^2,{x,0,nn}],x],{n,1,10}]],1]] //Grid (* _Geoffrey Critzer_, Sep 27 2013 *)

%Y For another version see A115720. Row lengths A000196.

%Y Cf. A115995, A115721, A115722, A008284, A006918.

%K nonn,tabf,changed

%O 1,2

%A _Emeric Deutsch_, Feb 11 2006

%E Edited and verified by _Franklin T. Adams-Watters_ Mar 11 2006

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Last modified January 29 17:35 EST 2015. Contains 253887 sequences.