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A088714
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G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)).
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10
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1, 1, 3, 13, 69, 419, 2809, 20353, 157199, 1281993, 10963825, 97828031, 907177801, 8716049417, 86553001779, 886573220093, 9351927111901, 101447092428243, 1130357986741545, 12923637003161409, 151479552582252239
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| G.f.: A(x) = f(x*A(x)) = (1-1/f(x))/x where f(x) is the g.f. of A088713.
Given g.f. A(x), then B(x)=x*A(x) satisfies 0=f(x, B(x), B(B(x))) where f(a0, a1, a2)=a0-a1+a1*a2 . - Michael Somos May 21 2005
G.f. satisfies: A(x) = 1/(1 - x*A(x)*A(x*A(x))).
G.f. satisfies: A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(n+k+m,k)/(n+k+m) * a(n-k,k).
(End)
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)]/(n+1)! ). [Paul D. Hanna, Dec 17 2010]
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EXAMPLE
| Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + xAB;
B = A + xBC;
C = B + xCD;
D = C + xDE;
E = D + xEF ; ...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x) + [d/dx x^3*A(x)^2]/2! + [d^2/dx^2 x^5*A(x)^3]/3! + [d^3/dx^3 x^7*A(x)^4]/4! + [d^4/dx^4 x^9*A(x)^5]/5! +...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, A))); polcoeff(A, n))} /* Michael Somos May 21 2005 */
(PARI) {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=1, n, A=1/(1-x*A*subst(A, x, x*A))); polcoeff(A, n))}
(PARI) {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=0, n, A=(1/x)*serreverse(x-x^2*A)); polcoeff(A, n))}
(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+k+m, k)/(n+k+m)*a(n-k, k))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]
(PARI) /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=0, n, Dx(m, x^(2*m+1)*A^(m+1))/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
| Cf. A088713.
Apart from signs, same as A067145. - Philippe DELEHAM, Jun 18 2006
Cf. A002449, A030266, A087949, A088717, A091713, A120971.
Sequence in context: A074534 A153395 A067145 * A192739 A088368 A196794
Adjacent sequences: A088711 A088712 A088713 * A088715 A088716 A088717
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 12 2003, May 22 2008
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