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A143546 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(-x)^2. 9
1, 1, 1, 3, 5, 18, 35, 136, 285, 1155, 2530, 10530, 23751, 100688, 231880, 996336, 2330445, 10116873, 23950355, 104819165, 250543370, 1103722620, 2658968130, 11777187240, 28558343775, 127067830773, 309831575760, 1383914371728 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..27.

Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1. - From N. J. A. Sloane, Jul 12 2011

FORMULA

G.f.: A(x) = G(x^2) + x*G(x^2)^3 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

a(2n) = C(5n,n)/(4n+1); a(2n+1) = C(5n+2,n)*3/(4n+3).

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 18*x^5 + 35*x^6 + 136*x^7 +...

A(x) = 1 + x*A(x)^3*A(-x)^2 where

A(x)^3 = 1 + 3x + 6x^2 + 16x^3 + 39x^4 + 114x^5 + 304x^6 + 936x^7 +...

A(-x)^2 = 1 - 2x + 3x^2 - 8x^3 + 17x^4 - 52x^5 + 125x^6 - 408x^7 +...

Also, A(x) = G(x^2) + x*G(x^2)^3 where

G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...

G(x)^3 = 1 + 3*x + 18*x^2 + 136*x^3 + 1155*x^4 + 10530*x^5 +...

MATHEMATICA

terms = 28;

A[_] = 1; Do[A[x_] = 1 + x A[x]^3 A[-x]^2 + O[x]^terms // Normal, {terms}];

CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

PROG

(PARI) {a(n)=local(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^3*subst(A^2, x, -x)); polcoeff(A, n)}

(PARI) {a(n)=local(m=n\2, p=2*(n%2)+1); binomial(5*m+p-1, m)*p/(4*m+p)}

CROSSREFS

Cf. A002294, A047749, A118970.

Sequence in context: A039584 A191717 A136131 * A069066 A011964 A123793

Adjacent sequences:  A143543 A143544 A143545 * A143547 A143548 A143549

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 23 2008

STATUS

approved

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Last modified May 12 13:46 EDT 2021. Contains 343823 sequences. (Running on oeis4.)