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

%I A104383 #21 Feb 16 2025 08:32:56
%S A104383 1,1,2,4,10,27,76,222,668,2048,6378,20132,64234,206848,671418,2194432,
%T A104383 7215644,23853318,79229676,264288462,884987529,2973772212,10024300890,
%U A104383 33888946600,114872472064,390334057172,1329347719190,4536808055808,15513418629884
%N A104383 Number of distinct partitions of triangular numbers n*(n+1)/2.
%C A104383 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).
%D A104383 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.
%H A104383 G. C. Greubel, <a href="/A104383/b104383.txt">Table of n, a(n) for n = 0..1000</a>
%H A104383 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A104383 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q.</a>
%F A104383 Limit_{n-->inf} a(n+1)/a(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...
%F A104383 a(n) = A000009(A000217(n)). - _Alois P. Heinz_, Nov 24 2016
%p A104383 with(numtheory):
%p A104383 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A104383       `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
%p A104383     end:
%p A104383 a:= n-> b(n*(n+1)/2):
%p A104383 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 24 2016
%t A104383 Join[{1},PartitionsQ/@Accumulate[Range[30]]] (* _Harvey P. Dale_, Dec 29 2012 *)
%o A104383 (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)}
%Y A104383 Cf. A000009, A000217, A066655, A104382.
%K A104383 nonn
%O A104383 0,3
%A A104383 _Paul D. Hanna_, Mar 04 2005
%E A104383 a(0)=1 prepended by _Alois P. Heinz_, Aug 05 2016