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A088714 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x*A(x)). 16
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; text; internal format)
OFFSET

0,3

COMMENTS

Equals row sums of triangle A291820.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

G.f. satisfies:

(1) A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).

(2) A(x) = 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n!.

(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n/n! ).

(4) A(x) = 1/(1 - x*A(x)*A(x*A(x))).

(5) 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

From Paul D. Hanna, 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)

a(n) = Sum_{k=0..n} A291820(n+1,k). - Paul D. Hanna, Sep 01 2017

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...

The g.f. A(x) satisfies:

x*A(x) = x + x^2*A(x) + d/dx x^4*A(x)^2/2! + d^2/dx^2 x^6*A(x)^3/3! + d^3/dx^3 x^8*A(x)^4/4! +...

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! +...

From Paul D. Hanna, Apr 16 2007: (Start)

G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:

A = 1 + x*A*B;

B = A + x*B*C;

C = B + x*C*D;

D = C + x*D*E;

E = D + x*E*F ; ...

where variables B,C,D,E,..., are formed from successive iterations of x*A(x):

B = A(x)*A(x*A(x)), C = B*A(x*B), D = C*A(x*C), E = D*A(x*D), ...;

more explicilty,

B = 1 + 2*x + 8*x^2 + 42*x^3 + 258*x^4 + 1764*x^5 + 13070*x^6 +...,

C = 1 + 3*x + 15*x^2 + 93*x^3 + 655*x^4 + 5039*x^5 + 41453*x^6 +...,

D = 1 + 4*x + 24*x^2 + 172*x^3 + 1372*x^4 + 11796*x^5 +...,

E = 1 + 5*x + 35*x^2 + 285*x^3 + 2545*x^4 + 24255*x^5 +...,

... (End)

Related expansions:

A(x*A(x)) = 1 + x + 4*x^2 + 22*x^3 + 142*x^4 + 1016*x^5 + 7838*x^6 + 64174*x^7 + 552112*x^8 +...

A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1054*x^5 + 7039*x^6 + 50632*x^7 + 387613*x^8 +...

d/dx x^4*A(x)^2/2! = 2*x^3 + 5*x^4 + 21*x^5 + 112*x^6 + 692*x^7 + 4743*x^8 +...

d^2/dx^2 x^6*A(x)^3/3! = 5*x^4 + 21*x^5 + 112*x^6 + 696*x^7 + 4815*x^8 +...

d^3/dx^3 x^8*A(x)^4/4! = 14*x^5 + 84*x^6 + 540*x^7 + 3795*x^8 +...

d^4/dx^4 x^10*A(x)^5/5! = 42*x^6 + 330*x^7 + 2475*x^8 + 19305*x^9 +...

...

d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n! = A000108(n)*x^(n+1) +...

MATHEMATICA

m = 21; A[_] = 1; Do[A[x_] = 1 + x A[x]^2 A[x A[x]] + O[x]^m, {m}];

CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)

PROG

(PARI) {a(n) = my(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))))} \\ Paul D. Hanna, 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=1, n, Dx(m-1, x^(2*m-1)*A^m/m!))+x*O(x^n))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* n-th Derivative: */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

/* G.f.: [Paul D. Hanna, May 31 2012] */

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+(1/x)*sum(m=1, n+1, Dx(m-1, x^(2*m)*A^m/m!))+x*O(x^n)); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A291820, A088713, A212910, A212919.

Apart from signs, same as A067145. - Philippe Deléham, Jun 18 2006

Cf. A002449, A030266, A087949, A088717, A091713, A120971.

Sequence in context: A074534 A153395 A243688 * A067145 A192739 A088368

Adjacent sequences:  A088711 A088712 A088713 * A088715 A088716 A088717

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 12 2003, May 22 2008

STATUS

approved

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Last modified June 23 09:07 EDT 2021. Contains 345397 sequences. (Running on oeis4.)