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A178927 Number of partitions into a triangular number of parts.

%I A178927 #33 Nov 11 2020 07:57:13
%S A178927 1,1,1,2,2,3,5,6,8,11,15,19,26,32,42,54,69,86,111,137,173,215,268,329,
%T A178927 409,499,614,748,914,1106,1346,1621,1958,2352,2827,3380,4048,4821,
%U A178927 5746,6824,8102,9587,11346,13383,15781,18566,21824,25597,30007,35100,41029
%N A178927 Number of partitions into a triangular number of parts.
%H A178927 Vaclav Kotesovec, <a href="/A178927/b178927.txt">Table of n, a(n) for n = 0..8000</a> (terms 0..2000 from Alois P. Heinz)
%F A178927 G.f.: Sum_{i>=0} x^(i*(i+1)/2) / Product_{j=1..i*(i+1)/2} (1 - x^j). - _Ilya Gutkovskiy_, May 07 2017
%e A178927 For example there are 7 unrestricted partitions of 5, namely: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1 and 1+1+1+1+1. Of these we allow only those with 1,3,6,10,... parts.  These are 5, 3+1+1 and 2+2+1. So a(5)=3.
%p A178927 b:= proc(n, i, t) option remember; `if`(n=0 or i=1, `if`(issqr(
%p A178927       1+8*(t+n)), 1, 0), b(n, i-1, t)+b(n-i, min(i, n-i), t+1))
%p A178927     end:
%p A178927 a:= n-> b(n$2, 0):
%p A178927 seq(a(n), n=0..80);  # _Alois P. Heinz_, Jul 29 2017
%t A178927 b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, If[Sqrt[1 + 8(t + n)] // IntegerQ, 1, 0], b[n, i - 1, t] + b[n - i, Min[i, n - i], t + 1]];
%t A178927 a[n_] := b[n, n, 0];
%t A178927 a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 11 2020, after _Alois P. Heinz_ *)
%o A178927 (Sage) A178927 = lambda n: 1 if n == 0 else sum(number_of_partitions(n,k=tri) for tri in [1..n] if is_triangular_number(tri)) # [_D. S. McNeil_, Dec 30 2010]
%Y A178927 Cf. A007294.
%K A178927 nonn,easy
%O A178927 0,4
%A A178927 _David S. Newman_, Dec 29 2010