A206290 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).

1, 1, 2, 3, 5, 7, 12, 17, 29, 44, 77, 114, 218, 330, 617, 987, 1913, 2968, 6068, 9500, 19263, 31399, 64268, 101702, 218891, 348559, 735823, 1205239, 2576727, 4119884, 9100854, 14588992, 31841260, 52163378, 114485092, 183947681, 414704366, 667453931, 1487920000

Offset

0,3

Comments

Compare to the g.f. of partition numbers (A000041): Sum_{n>=0} Product_{k=1..n} x/(1 - x^k).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

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,

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 12*x^6 + 17*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 + x^3 + x^4 + x^5 + x^6 +...+ x^(n+1) +...
G_2(x) = x + x^3 + 2*x^5 + 5*x^7 + 14*x^9 +...+ A000108(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 3*x^7 + 12*x^10 + 55*x^13 +...+ A001764(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 4*x^9 + 22*x^13 + 140*x^17 +...+ A002293(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 5*x^11 + 35*x^16 + 285*x^21 +...+ A002294(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 6*x^13 + 51*x^19 + 506*x^25 +...+ A002295(n)*x^(6*n+1) +...
G_7(x) = x + x^8 + 7*x^15 + 70*x^22 + 819*x^29 +...+ A002296(n)*x^(7*n+1) +...
Note that G_n(x) = x + x*G_n(x)^n.

Prog

(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, 45, print1(a(n), ", "))

Crossrefs

Cf. A206289, A194560, A110448.

Sequence in context: A334271 A206788 A002965 * A091696 A334683 A280303

Adjacent sequences: A206287 A206288 A206289 * A206291 A206292 A206293

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2012

Status

approved