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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A347147 A295213 A118245 * A086629 A203948 A296990
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 04 2005
STATUS
approved

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Last modified June 20 07:26 EDT 2024. Contains 373512 sequences. (Running on oeis4.)