A271867 G.f. A(x) satisfies: A(x) = x + A( x^2 + x*A(x)^2 ).

1, 1, 1, 3, 5, 14, 30, 82, 203, 552, 1458, 4004, 10956, 30514, 85259, 240507, 681571, 1943472, 5565744, 16011492, 46233297, 133975566, 389455910, 1135431759, 3319060758, 9726061473, 28565447104, 84073146827, 247924840773, 732439856638, 2167507140543, 6424491527538, 19070573498367, 56688719414910, 168733726744153, 502859937709589, 1500383417733522, 4481672952197057, 13400947416395067, 40111136395590224

Offset

1,4

Comments

Compare g.f. to: C(x) = x + C( x^2 + 2*x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) ~ c * d^n / n^(3/2), where d = 3.109781515236165... and c = 0.1963355843719... . - Vaclav Kotesovec, Apr 16 2016

Example

G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 14*x^6 + 30*x^7 + 82*x^8 + 203*x^9 + 552*x^10 + 1458*x^11 + 4004*x^12 + 10956*x^13 + 30514*x^14 + 85259*x^15 +...
where A(x) = x + A( x^2 + x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 17*x^6 + 44*x^7 + 107*x^8 + 282*x^9 + 739*x^10 + 1994*x^11 + 5414*x^12 + 14906*x^13 + 41374*x^14 + 115820*x^15 +...
Let G(x,y) = x + G(x^2 + x*y*G(x,y)^2, y), then the coefficient of x^n in G(x,y) begins:
n=1: 1;
n=2: 1;
n=3: y;
n=4: 1 + 2*y;
n=5: 3*y + 2*y^2;
n=6: 7*y + 7*y^2;
n=7: 4*y + 21*y^2 + 5*y^3;
n=8: 1 + 6*y + 46*y^2 + 29*y^3;
n=9: 9*y + 65*y^2 + 114*y^3 + 15*y^4;
n=10: 13*y + 113*y^2 + 304*y^3 + 122*y^4;
n=11: 8*y + 169*y^2 + 649*y^3 + 582*y^4 + 50*y^5;
n=12: 19*y + 229*y^2 + 1311*y^3 + 1931*y^4 + 514*y^5;
n=13: 14*y + 326*y^2 + 2289*y^3 + 5235*y^4 + 2915*y^5 + 177*y^6;
n=14: 4*y + 511*y^2 + 3800*y^3 + 12353*y^4 + 11667*y^5 + 2179*y^6;
n=15: 8*y + 528*y^2 + 6365*y^3 + 25663*y^4 + 37605*y^5 + 14439*y^6 + 651*y^7;
n=16: 1 + 14*y + 602*y^2 + 9933*y^3 + 50117*y^4 + 102960*y^5 + 67567*y^6 + 9313*y^7; ...
where the coefficients of x^n at y=2 yield the Catalan sequence (A000108)
and the coefficients of x^n at y=1 yield this sequence.

Prog

(PARI) {a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + subst(A, x, x^2 + x*A^2) ) ; polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A271868.

Sequence in context: A192478 A198785 A222380 * A295064 A052974 A284415

Adjacent sequences: A271864 A271865 A271866 * A271868 A271869 A271870

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2016

Status

approved