A330376 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

1, 3, 6, 10, 2, 15, 5, 21, 14, 28, 26, 36, 50, 45, 80, 3, 55, 130, 7, 66, 190, 19, 78, 280, 41, 91, 385, 80, 105, 532, 143, 120, 700, 248, 136, 924, 399, 4, 153, 1176, 627, 9, 171, 1500, 949, 24, 190, 1860, 1397, 51, 210, 2310, 2003, 107, 231, 2805, 2823, 193

Offset: 1

Author

Omar E. Pol, Dec 22 2019

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   1;
   3;
   6;
  10,  2;
  15,  5;
  21, 14;
  28, 26;
  36, 50;
  45, 80, 3;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1799 (rows 1..200)
• Eric Weisstein's World of Mathematics, Durfee Square

Crossrefs

Row sums give A006128, n >= 1.

Column 1 gives A000217, n >= 1.

Cf. A114089, A115721, A115722, A115994, A116858, A118198, A208474, A330369, A330640, A330641, A330642, A330643.

Cf. A330369.

Programs

PARI

\\ by enumeration over partitions.
ds(p)={for(i=2, #p, if(p[#p+1-i]Andrew Howroyd, Feb 02 2022

PARI

\\ by generating function.
P(n,k,y)={1/prod(j=1, k, 1 - y*x^j + O(x*x^n))}
T(n,k)={my(r=n-k^2); if(r<0, 0, subst(deriv(polcoef(y^k*P(r,k,1)*P(r,k,y), r)), y, 1))}
{ for(n=1, 10, print(vector(sqrtint(n), k, T(n,k)))) } \\ Andrew Howroyd, Feb 02 2022

Extensions

Terms a(10) and beyond from Andrew Howroyd, Feb 02 2022