A190271 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^A001952(n), where A001952(n) = [n*(2+sqrt(2))].

1, 1, 4, 22, 141, 986, 7295, 56145, 444900, 3605429, 29742671, 248933630, 2108590305, 18042013096, 155711676129, 1353913728412, 11849013783676, 104293377329508, 922643326714355, 8199321046129671, 73162599757076951

Offset

0,3

Comments

Compare to the g.f. of A190270, F(x), which satisfies:

* F(x) = Sum_{n>=0} x^n*F(x)^A001951(n),

where A001951 is the complementary Beatty sequence to A001952.

Links

Table of n, a(n) for n=0..20.

Formula

G.f. satisfies: A(x) = F(x*A(x)^2) where A(x/F(x)^2) = F(x) is the g.f. of A190270, which in turn satisfies: F(x) = Sum_{n>=0} x^n*F(x)^[n*sqrt(2)].

G.f.: A(x) = sqrt((1/x)*Series_Reversion(x/F(x)^2)) where F(x) is the g.f. of A190270.

Example

 G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 141*x^4 + 986*x^5 + 7295*x^6 +...
The g.f. satisfies:
F(x) = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^10 + x^4*A(x)^13 + x^5*A(x)^17 + x^6*A(x)^20 + x^7*A(x)^23 +...+ x^n*A(x)^A001952(n) +...
The g.f. of A190270, F(x) = A(x/F(x)^2), satisfies:
F(x) = 1 + x*F(x) + x^2*F(x)^2 + x^3*F(x)^4 + x^4*F(x)^5 + x^5*F(x)^7 + x^6*F(x)^8 + x^7*F(x)^9 + x^8*F(x)^11 +...+ x^n*F(x)^A001951(n) +...
and begins:
F(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 49*x^5 + 169*x^6 + 605*x^7 +...
Since A(x) = F(x*A(x)^2), then:
A(x) = 1 + x*A(x)^2 + 2*x^2*A(x)^4 + 5*x^3*A(x)^6 + 15*x^4*A(x)^8 +...

Prog

 (PARI) {a(n)=local(A=1+x, t=sqrt(2)-1); for(i=1, n, A=sum(m=0, n, x^m*(A+x*O(x^n))^floor(m+m/t))); polcoeff(A, n)}

Crossrefs

Cf. A190270, A001952; variant: A186577.

Sequence in context: A077056 A227404 A366677 * A045744 A369504 A243626

Adjacent sequences: A190268 A190269 A190270 * A190272 A190273 A190274

Keyword

nonn

Author

Paul D. Hanna, May 06 2011

Status

approved