OFFSET
0,4
COMMENTS
Also number of partitions into non-pronic numbers (cannot be written as i*(i+1)).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 129 terms from Reinhard Zumkeller)
Mircea Merca and Emil Simion, n-Color Partitions into Distinct Parts as Sums over Partitions, Symmetry (2023) Vol. 15, Iss. 11.
FORMULA
G.f.: Product_{i>=1} (1-x^(i*(i+1)))/(1-x^i).
G.f.: (1+x) * (1+x^2+x^4) * (1+x^3+x^6+x^9) * (1+x^4+x^8+x^12+x^16) * ... (g.f. above, expanded). - Joerg Arndt, Apr 01 2014
G.f.: Product_{n>=1} (1 - q^(n*(n+1))) / Product_{n>=1} (1 - q^n). - Joerg Arndt, Apr 01 2014
a(n) = p(n,1,1) with p(n,t,k) = if t<0 then 0 else if k<=n then p(n-k,t-1,k)+p(n,k+1,k+1) else 0^n. - Reinhard Zumkeller, Jan 20 2010
a(n) ~ exp(Pi*sqrt(2*n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / 2^(3/4) - 3*Zeta(3/2)^2/(32*Pi)) / sqrt(2*n). - Vaclav Kotesovec, Jan 01 2017
EXAMPLE
a(5)=4 because we have [5], [4,1], [3,2] and [2,2,1] ([3,1,1], [2,1,1,1] and [1,1,1,1,1] do not qualify).
MAPLE
g:=product((1-x^(j*(j+1)))/(1-x^j), j=1..53): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..49); # Emeric Deutsch, Mar 04 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
`if`(issqr(4*d+1), 0, d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..80); # Alois P. Heinz, Apr 01 2014
MATHEMATICA
CoefficientList[Series[Product[Sum[x^(i j ), {i, 0, j}], {j, 1, 49}], {x, 0, 49}], x]
(* Second program: *)
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[IntegerQ @ Sqrt[4*d+1], 0, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 30 2017, after Alois P. Heinz *)
PROG
(PARI) N=66; q='q+O('q^N); Vec( prod(n=1, N, sum(k=0, n, q^(k*n)) ) ) \\ Joerg Arndt, Apr 01 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Dec 19 1999
STATUS
approved