A271843 G.f. A(x) satisfies: A(x) = x + A( x*A(x) + x*A(x)^3 ).

1, 1, 1, 3, 8, 20, 55, 159, 464, 1383, 4200, 12910, 40112, 125832, 397888, 1266848, 4058263, 13070453, 42297553, 137467673, 448499679, 1468388784, 4822816903, 15886282268, 52468807845, 173718343364, 576466929104, 1916968549390, 6387086400663, 21319636605919, 71284279000874, 238724756808108, 800659887614429, 2689098091847122

Offset

1,4

Comments

Compare g.f. to: C(x) = x + C( x*C(x) + 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 = 3.51361449558530219727... and c = 0.1466906366440109... . - Vaclav Kotesovec, Apr 16 2016

Example

G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 8*x^5 + 20*x^6 + 55*x^7 + 159*x^8 + 464*x^9 + 1383*x^10 + 4200*x^11 + 12910*x^12 +...
where A(x) = x + A( x*A(x) + x*A(x)^3 ).
RELATED SERIES.
A(x) + A(x)^3 = x + x^2 + 2*x^3 + 6*x^4 + 14*x^5 + 36*x^6 + 103*x^7 + 297*x^8 + 867*x^9 + 2598*x^10 + 7908*x^11 + 24337*x^12 + 75725*x^13 + 237822*x^14
+...

Prog

(PARI) {a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + subst(A, x, x*A + x*A^3) ) ; polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A230953 A026995 A018035 * A122228 A018790 A268042

Adjacent sequences: A271840 A271841 A271842 * A271844 A271845 A271846

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2016

Status

approved