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

1,5

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...

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.

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)

Example

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; ...

Prog

(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))
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print(""))

Crossrefs

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

Sequence in context: A347147 A295213 A118245 * A086629 A203948 A296990

Adjacent sequences: A104379 A104380 A104381 * A104383 A104384 A104385

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 04 2005

Status

approved