A116858 -id:A116858 - cp's OEIS frontend

A116859 Sum of the sizes of the Durfee squares of all partitions of n into distinct parts.

1, 1, 2, 2, 4, 6, 8, 10, 14, 18, 22, 29, 36, 46, 59, 72, 88, 110, 132, 160, 194, 232, 276, 330, 392, 464, 550, 648, 760, 894, 1044, 1216, 1418, 1644, 1905, 2204, 2540, 2924, 3364, 3859, 4420, 5060, 5778, 6590, 7514, 8544, 9706, 11018, 12484, 14130, 15980

Offset: 1

Emeric Deutsch, Feb 26 2006

nonn

a(n)=Sum(k*A116858(n,k),k>=1).

			a(8)=10 because the partitions of 8 into distinct parts are [8],[7,1],[6,2],[5,3],[5,2,1] and [4,3,1], the sum of the sizes of their Durfee squares being 1+1+2+2+2+2=10.

Cf. A116858.

f:=sum(k*x^(k*(3*k-1)/2)*(1+x^(2*k))*product((1+x^j)/(1-x^j),j=1..k-1)/(1-x^k),k=1..10): fser:=series(f,x=0,60): seq(coeff(fser,x^n),n=1..55);

G.f.=sum(kx^(k(3k-1)/2)*(1+x^(2k))*product((1+x^j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity).

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))).

Original entry on oeis.org

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

Omar E. Pol, Dec 22 2019

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

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

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.

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

\\ 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

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

cp's OEIS Frontend

A116859 Sum of the sizes of the Durfee squares of all partitions of n into distinct parts.

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