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A104383 Number of distinct partitions of triangular numbers n*(n+1)/2. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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
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
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
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 *)
PROG
(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)}
CROSSREFS
Sequence in context: A216434 A220829 A339838 * A205480 A108523 A157003
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 04 2005
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Aug 05 2016
STATUS
approved

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Last modified May 23 07:28 EDT 2024. Contains 372760 sequences. (Running on oeis4.)