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A162162 G.f. satisfies: A(x) = Catalan(x + x^2 + x^3*A(x)) where Catalan(x) = (1-sqrt(1-4*x))/(2x) is the g.f. of A000108. 2
1, 1, 3, 10, 36, 139, 560, 2328, 9914, 43027, 189619, 846267, 3817105, 17373048, 79687447, 367991891, 1709477714, 7983062151, 37454903501, 176470241003, 834601583199, 3960757007408, 18855383609076, 90019104197240 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

FORMULA

O.g.f.: A(x) = 1 + (x+x^2)*A(x)^2 + x^3*A(x)^3 [From Simon Plouffe].

a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-2*k+j+1,n-k)/(2*n-2*k+j+m) * C(n-k,k-j)*C(k-j,j).

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

a(n,m) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-2*k+j+m,n-k)*m/(2*n-2*k+j+m) * C(n-k,k-j)*C(k-j,j).

a(n) ~ s * sqrt((1 - r*s)/(Pi*(1 - r*s*(1 + r)))) / (2*n^(3/2)*r^n), where r = 0.1964570949596604593878365861880061480194047064202... and s = 1.944271909999158785636694674925104941762473438446... are real roots of the system of equations 1 + r*(1 + r)*s^2 + r^3*s^3 = s, r*s*(2 + 2*r + 3*r^2*s) = 1. - Vaclav Kotesovec, Nov 22 2017

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + 139*x^5 + 560*x^6 +...

A(x) = Catalan(x + x^2 + x^3*A(x)) where:

Catalan(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...

PROG

(PARI) {a(n, m=1)=sum(k=0, n, sum(j=0, k, binomial(2*n-2*k+j+m, n-k)*m/(2*n-2*k+j+m)*binomial(n-k, k-j)*binomial(k-j, j)))}

(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=2/(1+sqrt(1-4*(x+x^2 +x^3*A)))); polcoeff(A^m, n)}

(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+(x+x^2)*A^2+x^3*A^3); polcoeff(A^m, n)}

CROSSREFS

Cf. A000108.

Sequence in context: A149041 A202834 A129247 * A149042 A081921 A165792

Adjacent sequences:  A162159 A162160 A162161 * A162163 A162164 A162165

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 26 2009

STATUS

approved

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Last modified December 17 00:05 EST 2018. Contains 318191 sequences. (Running on oeis4.)