|
%I
%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).
%H Alois P. Heinz, <a href="/A115994/b115994.txt">Rows n = 1..620, flattened</a>
%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;
%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
%Y For another version see A115720. Row lengths A000196.
%Y Cf. A115995, A115721, A115722, A008284, A006918.
%K nonn,tabf
%O 1,2
%A _Emeric Deutsch_, Feb 11 2006
%E Edited and verified by _Franklin T. Adams-Watters_ Mar 11 2006
|
