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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
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
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1799 (rows 1..200)
Eric Weisstein's World of Mathematics, Durfee Square
EXAMPLE
Triangle begins:
1;
3;
6;
10, 2;
15, 5;
21, 14;
28, 26;
36, 50;
45, 80, 3;
PROG
(PARI) \\ by enumeration over partitions.
ds(p)={for(i=2, #p, if(p[#p+1-i]<i, return(i-1))); #p}
row(n)={my(v=vector(sqrtint(n))); forpart(p=n, v[ds(p)] += #p); v}
{ for(n=1, 10, print(row(n))) } \\ 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
CROSSREFS
Row sums give A006128, n >= 1.
Column 1 gives A000217, n >= 1.
Cf. A330369.
Sequence in context: A194047 A194035 A194049 * A120028 A333611 A329153
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 22 2019
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Feb 02 2022
STATUS
approved