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A116647 Triangle of number of partitions that fit in an n X n box (but not in an (n-1) X (n-1) box) with Durfee square k.

%I A116647 #9 Feb 16 2025 08:33:00
%S A116647 1,3,1,5,8,1,7,27,15,1,9,64,84,24,1,11,125,300,200,35,1,13,216,825,
%T A116647 1000,405,48,1,15,343,1911,3675,2695,735,63,1,17,512,3920,10976,12740,
%U A116647 6272,1232,80,1,19,729,7344,28224,47628,37044,13104,1944,99,1,21,1000,12825
%N A116647 Triangle of number of partitions that fit in an n X n box (but not in an (n-1) X (n-1) box) with Durfee square k.
%H A116647 G. C. Greubel, <a href="/A116647/b116647.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A116647 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square.</a>
%F A116647 T(n,k) = binomial(n,k)^2 - binomial(n-1,k)^2.
%e A116647 Triangle begins
%e A116647    1;
%e A116647    3,   1;
%e A116647    5,   8,   1;
%e A116647    7,  27,  15,   1;
%e A116647    9,  64,  84,  24,   1;
%e A116647   11, 125, 300, 200,  35,   1;
%t A116647 Table[Binomial[n, k]^2 - Binomial[n - 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Nov 20 2017 *)
%o A116647 (PARI) for(n=1,10, for(k=0,n, print1(binomial(n,k)^2 - binomial(n-1,k)^2, ", "))) \\ _G. C. Greubel_, Nov 20 2017
%Y A116647 Cf. A008459; row sums A051924.
%K A116647 easy,nonn,tabl
%O A116647 1,2
%A A116647 _Franklin T. Adams-Watters_, Feb 20 2006