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A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.

%I A104382 #28 Feb 16 2025 08:32:56
%S A104382 1,1,1,1,2,1,1,4,4,1,1,7,12,6,1,1,10,27,27,10,1,1,13,52,84,57,14,1,1,
%T A104382 17,91,206,221,110,21,1,1,22,147,441,674,532,201,29,1,1,27,225,864,
%U A104382 1747,1945,1175,352,41,1,1,32,331,1575,4033,5942,5102,2462,598,55,1,1,38,469
%N A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.
%C A104382 Secondary diagonal equals partitions of n - 1 (A000065).
%C A104382 Third diagonal is A104384.
%C A104382 Third column is A104385.
%C A104382 Row sums are A104383 where limit_{n --> inf} A104383(n+1)/A104383(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...
%D A104382 Abramowitz, M. and Stegun, I. A. (Editors). "Partitions into Distinct Parts." S24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, pp. 825-826, 1972.
%H A104382 Alois P. Heinz, <a href="/A104382/b104382.txt">Rows n = 1..55, flattened</a>
%H A104382 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A104382 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q.</a>
%F A104382 T(n, 1) = T(n, n) = 1.
%F A104382 T(n, n-1) = A000065(n).
%F A104382 T(n, 2) = [(n*(n+1)/2-1)/2].
%F A104382 From _Álvar Ibeas_, Jul 23 2020: (Start)
%F A104382 T(n, k) = A008284((n-k+1)*(n+k)/2, k).
%F A104382 T(n, k) = A026820((n-k)*(n+k+1)/2, k), with A026820(0, k) = 1. (End)
%e A104382 Rows begin:
%e A104382 1;
%e A104382 1, 1;
%e A104382 1, 2, 1;
%e A104382 1, 4, 4, 1;
%e A104382 1, 7, 12, 6, 1;
%e A104382 1, 10, 27, 27, 10, 1;
%e A104382 1, 13, 52, 84, 57, 14, 1;
%e A104382 1, 17, 91, 206, 221, 110, 21, 1;
%e A104382 1, 22, 147, 441, 674, 532, 201, 29, 1;
%e A104382 1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1;
%e A104382 1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1; ...
%o A104382 (PARI) T(n,k)=if(n<k || k<1,0,polcoeff(polcoeff( prod(i=1,n*(n+1)/2,1+y*x^i,1+x*O(x^(n*(n+1)/2))),n*(n+1)/2,x),k,y))
%o A104382 for(n=1,12,for(k=1,n,print1(T(n,k),", "));print(""))
%Y A104382 Cf. A008289, A000009, A000065, A104383, A104384, A104385.
%K A104382 nonn,tabl
%O A104382 1,5
%A A104382 _Paul D. Hanna_, Mar 04 2005