A104383 Number of distinct partitions of triangular numbers n*(n+1)/2.

1, 1, 2, 4, 10, 27, 76, 222, 668, 2048, 6378, 20132, 64234, 206848, 671418, 2194432, 7215644, 23853318, 79229676, 264288462, 884987529, 2973772212, 10024300890, 33888946600, 114872472064, 390334057172, 1329347719190, 4536808055808, 15513418629884

Offset: 0

Author

Paul D. Hanna, Mar 04 2005

Keywords

• nonn

Comments

Equals row sums of triangle A104382. Asymptotics: a(n) ~ exp(Pi*sqrt((n^2+n)/6))/(2*6^(1/4))/(n^2+n)^(3/4).

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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. A000009, A000217, A066655, A104382.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n*(n+1)/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 24 2016

Mathematica

Join[{1},PartitionsQ/@Accumulate[Range[30]]] (* Harvey P. Dale, Dec 29 2012 *)

PARI

{a(n)=polcoeff(prod(k=1,n*(n+1)/2,1+x^k,1+x*O(x^(n*(n+1)/2))),n*(n+1)/2)}

Formula

Limit_{n-->inf} a(n+1)/a(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...

a(n) = A000009(A000217(n)). - Alois P. Heinz, Nov 24 2016

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 05 2016