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.

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, 17, 91, 206, 221, 110, 21, 1, 1, 22, 147, 441, 674, 532, 201, 29, 1, 1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1, 1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1, 1, 38, 469

Offset: 1

Author

Paul D. Hanna, Mar 04 2005

Keywords

• nonn
• tabl

Comments

Secondary diagonal equals partitions of n - 1 (A000065).

Third diagonal is A104384.

Third column is A104385.

Row sums are A104383 where limit_{n --> inf} A104383(n+1)/A104383(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...

Examples

			Rows begin:
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, 17, 91, 206, 221, 110, 21, 1;
1, 22, 147, 441, 674, 532, 201, 29, 1;
1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1;
1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1; ...
		

References

• 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.

Links

• Alois P. Heinz, Rows n = 1..55, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Eric Weisstein's World of Mathematics, Partition Function Q.

Crossrefs

Cf. A008289, A000009, A000065, A104383, A104384, A104385.

Programs

PARI

T(n,k)=if(n
				

Formula

T(n, 1) = T(n, n) = 1.

T(n, n-1) = A000065(n).

T(n, 2) = [(n*(n+1)/2-1)/2].

From Álvar Ibeas, Jul 23 2020: (Start)

T(n, k) = A008284((n-k+1)*(n+k)/2, k).

T(n, k) = A026820((n-k)*(n+k+1)/2, k), with A026820(0, k) = 1. (End)