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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))).

%I A330376 #22 Feb 16 2025 08:33:59
%S A330376 1,3,6,10,2,15,5,21,14,28,26,36,50,45,80,3,55,130,7,66,190,19,78,280,
%T A330376 41,91,385,80,105,532,143,120,700,248,136,924,399,4,153,1176,627,9,
%U A330376 171,1500,949,24,190,1860,1397,51,210,2310,2003,107,231,2805,2823,193
%N 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))).
%H A330376 Andrew Howroyd, <a href="/A330376/b330376.txt">Table of n, a(n) for n = 1..1799</a> (rows 1..200)
%H A330376 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square</a>
%e A330376 Triangle begins:
%e A330376    1;
%e A330376    3;
%e A330376    6;
%e A330376   10,  2;
%e A330376   15,  5;
%e A330376   21, 14;
%e A330376   28, 26;
%e A330376   36, 50;
%e A330376   45, 80, 3;
%o A330376 (PARI) \\ by enumeration over partitions.
%o A330376 ds(p)={for(i=2, #p, if(p[#p+1-i]<i, return(i-1))); #p}
%o A330376 row(n)={my(v=vector(sqrtint(n))); forpart(p=n, v[ds(p)] += #p); v}
%o A330376 { for(n=1, 10, print(row(n))) } \\ _Andrew Howroyd_, Feb 02 2022
%o A330376 (PARI) \\ by generating function.
%o A330376 P(n,k,y)={1/prod(j=1, k, 1 - y*x^j + O(x*x^n))}
%o A330376 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))}
%o A330376 { for(n=1, 10, print(vector(sqrtint(n), k, T(n,k)))) } \\ _Andrew Howroyd_, Feb 02 2022
%Y A330376 Row sums give A006128, n >= 1.
%Y A330376 Column 1 gives A000217, n >= 1.
%Y A330376 Cf. A114089, A115721, A115722, A115994, A116858, A118198, A208474, A330369, A330640, A330641, A330642, A330643.
%Y A330376 Cf. A330369.
%K A330376 nonn,tabf
%O A330376 1,2
%A A330376 _Omar E. Pol_, Dec 22 2019
%E A330376 Terms a(10) and beyond from _Andrew Howroyd_, Feb 02 2022