A190270 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^A001951(n), where A001951(n) = [n*sqrt(2)].

1, 1, 2, 5, 15, 49, 169, 605, 2226, 8364, 31956, 123770, 484862, 1917800, 7648470, 30722318, 124180334, 504720369, 2061489396, 8457050387, 34831589583, 143972841512, 597034531410, 2483173470124, 10356092457386, 43298360910159

Offset

0,3

Comments

Compare to the g.f. of A190271, G(x), which satisfies:

* G(x) = Sum_{n>=0} x^n*G(x)^A001952(n),

where A001952 is the complementary Beatty sequence to A001951.

Links

Table of n, a(n) for n=0..25.

Formula

G.f. satisfies: A(x) = G(x/A(x)^2) where A(x*G(x)^2) = G(x) is the g.f. of A190271, which in turn satisfies: G(x) = Sum_{n>=0} x^n*G(x)^[n*(2+sqrt(2))].

G.f.: A(x) = sqrt(x/Series_Reversion(x*G(x)^2)) where G(x) is the g.f. of A190271.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 49*x^5 + 169*x^6 +...
The g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^4 + x^4*A(x)^5 + x^5*A(x)^7 + x^6*A(x)^8 + x^7*A(x)^9 + x^8*A(x)^11 +...+ x^n*A(x)^A001951(n) +...
The g.f. of A190271, G(x) = A(x*G(x)^2), satisfies:
G(x) = 1 + x*G(x)^3 + x^2*G(x)^6 + x^3*G(x)^10 + x^4*G(x)^13 + x^5*G(x)^17 + x^6*G(x)^20 + x^7*G(x)^23 +...+ x^n*G(x)^A001952(n) +...
and begins:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 141*x^4 + 986*x^5 + 7295*x^6 +...
Since A(x) = G(x/A(x)^2), then:
A(x) = 1 + x/A(x)^2 + 4*x^2/A(x)^4 + 22*x^3/A(x)^6 + 141*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. A190271, A001951; variant: A186576.

Sequence in context: A001892 A176025 A084082 * A364588 A341342 A149939

Adjacent sequences: A190267 A190268 A190269 * A190271 A190272 A190273

Keyword

nonn

Author

Paul D. Hanna, May 06 2011

Status

approved