A178927 - OEIS

%I #33 Nov 11 2020 07:57:13

%S 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 409,499,614,748,914,1106,1346,1621,1958,2352,2827,3380,4048,4821,

%U 5746,6824,8102,9587,11346,13383,15781,18566,21824,25597,30007,35100,41029

%N Number of partitions into a triangular number of parts.

%H 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 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 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 b:= proc(n, i, t) option remember; `if`(n=0 or i=1, `if`(issqr(

%p       1+8*(t+n)), 1, 0), b(n, i-1, t)+b(n-i, min(i, n-i), t+1))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Jul 29 2017

%t 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 a[n_] := b[n, n, 0];

%t a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 11 2020, after _Alois P. Heinz_ *)

%o (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 Cf. A007294.

%K nonn,easy

%O 0,4

%A _David S. Newman_, Dec 29 2010