1,2

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.

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

Successive columns approach closer and closer to A000712. - njas, Mar 10 2007

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

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

Eric Weisstein's World of Mathematics, Durfee Square.

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

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.

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.

Triangle starts:

1;

2;

3;

4,1;

5,2;

6,5;

7,8;

8,14;

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

For another version see A115720. Row lengths A000196.

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

Sequence in context: A083480 A023133 A026280 this_sequence A071437 A129709 A133108

Adjacent sequences: A115991 A115992 A115993 this_sequence A115995 A115996 A115997

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 11 2006

Edited and verified by Frank Adams-Watters (FrankTAW(AT)Netscape net) Mar 11 2006