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A215505 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)^3). 3
1, 1, 4, 25, 200, 1888, 20158, 237357, 3032188, 41554144, 605964370, 9345693140, 151727166822, 2583300560490, 45984983349166, 853637181574329, 16489023127843088, 330789284905928356, 6880312907650893934, 148151276593976715612, 3297947033016039111690 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

a(n) = coefficient of x^n in (1+x*A(x))^(3*n+1)/(3*n+1) where A(x) = Sum_{n=0} a(n)*x^n.

Recurrence:

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*binomial(3*n+m,k)/(3*n+m) * a(n-k,k).

G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)^2*F(x,n+1)) for n>0 with F(x,0)=1.

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 200*x^4 + 1888*x^5 + 20158*x^6 +...

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

A = 1 + x*A^2*B;

B = A*(1 + x*B^2*C);

C = B*(1 + x*C^2*D);

D = C*(1 + x*D^2*E);

E = D*(1 + x*E^2*F); ...

where B(x) = A(x)*A(x*A(x)^3), C(x) = A(x)*B(x*A(x)^3),  D(x) = A(x)*C(x*A(x)^3), ...

Expansions of a few of the functions described above begin:

B(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 998*x^4 + 11258*x^5 + 139398*x^6 +...

C(x) = 1 + 3*x + 24*x^2 + 253*x^3 + 3090*x^4 + 41646*x^5 + 604636*x^6 +...

D(x) = 1 + 4*x + 40*x^2 + 512*x^3 + 7452*x^4 + 118016*x^5 + 1990284*x^6 +...

E(x) = 1 + 5*x + 60*x^2 + 905*x^3 + 15340*x^4 + 280400*x^5 + 5417498*x^6 +...

ALTERNATE GENERATING METHOD.

Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:

A = 1 + x*A^3*B,

B = 1 + x*(A*B)^3*C,

C = 1 + x*(A*B*C)^3*D,

D = 1 + x*(A*B*C*D)^3*E, etc.,

then B(x) = A(x*A(x)^3), C(x) = B(x*A(x)^3), D(x) = C(x*A(x)^3), etc.,

where A(x) = 1 + x*A(x)^3*A(x*A(x)^3) is the g.f. of this sequence.

Expansions of a few of the functions described above begin:

B(x) = 1 + x + 7*x^2 + 64*x^3 + 681*x^4 + 8058*x^5 + 103570*x^6 +...

C(x) = 1 + x + 10*x^2 + 121*x^3 + 1630*x^4 + 23678*x^5 + 364984*x^6 +...

D(x) = 1 + x + 13*x^2 + 196*x^3 + 3209*x^4 + 55660*x^5 + 1010248*x^6 +...

E(x) = 1 + x + 16*x^2 + 289*x^3 + 5580*x^4 + 112860*x^5 + 2367358*x^6 +...

PROG

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

(PARI) /* a(n) = [x^n] (1+x*A(x))^(3*n+1)/(3*n+1): */

{a(n)=local(A=1+x); for(i=0, n, A=sum(m=0, n, polcoeff((1+x*A+x*O(x^m))^(3*m+1)/(3*m+1), m)*x^m)+x*O(x^n)); polcoeff(A, n)}

(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(3*n+m, k)/(3*n+m)*a(n-k, k))))}

for(n=0, 31, print1(a(n), ", "))

CROSSREFS

Cf. A088717, A215506, A215507.

Sequence in context: A140094 A284859 A144647 * A182304 A060910 A195260

Adjacent sequences:  A215502 A215503 A215504 * A215506 A215507 A215508

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 13 2012

STATUS

approved

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Last modified June 24 00:11 EDT 2021. Contains 345403 sequences. (Running on oeis4.)