OFFSET
0,2
COMMENTS
Partial sums of A007294.
Number of partitions of n into triangular numbers if there are two kinds of 1's.
LINKS
FORMULA
G.f.: (1/(1 - x))*Sum_{j>=0} x^(j*(j+1)/2)/Product_{k=1..j} (1 - x^(k*(k+1)/2)).
From Vaclav Kotesovec, Apr 13 2018: (Start)
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2)^(1/3) / (2^(5/2) * sqrt(3) * Pi^(4/3) * n^(5/6)).
a(n) ~ 2 * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A007294(n). (End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
b(n, i-1)+(t->`if`(t>n, 0, b(n-t, i)))(i*(i+1)/2))
end:
a:= n-> b(n, isqrt(2*n)):
seq(a(n), n=0..100); # Alois P. Heinz, Apr 13 2018
MATHEMATICA
nmax = 55; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[1/(1 - x) Sum[x^(j (j + 1)/2)/Product[(1 - x^(k (k + 1)/2)), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 13 2018
STATUS
approved