A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).

1, 1, 2, 4, 10, 25, 73, 214, 679, 2189, 7331, 24867, 86269, 302144, 1072621, 3837768, 13853674, 50319789, 183941789, 675731105, 2494370326, 9244865453, 34394851701, 128390336942, 480749791772, 1805161153783, 6795744287172, 25643914891284, 96980809856731

Offset: 0

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

Comments

Compare to the g.f. of partitions of n into distinct parts (A000009): Sum_{n>=0} Product_{k=1..n} x*(1 + x^k).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x*(1 - x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
Note that G_n(x) = x + x*G_n(x)^(n+1).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Crossrefs

Cf. A206290, A194560.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x*(1-x^k+x*O(x^n))))),n)}
for(n=0,35,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:

(1) G_n(x) = Series_Reversion( x*(1 - x^n) ),

(2) G_n(x) = x + x*G_n(x)^(n+1),

(3) G_n(x) = Sum_{k>=0} binomial(n*k+k+1, k) * x^(n*k+1) / (n*k+k+1).

a(n) ~ c * 4^n / n^(3/2), where c = 0.19197348199... . - Vaclav Kotesovec, Nov 06 2014