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A215507 G.f. satisfies: A(x) = 1 + x*A(x)^5*A(x*A(x)^5). 3
1, 1, 6, 56, 661, 9141, 142522, 2448544, 45653707, 913964706, 19491269046, 440154262428, 10475920613965, 261802864005533, 6848792691398646, 187061918111607286, 5322557388634585229, 157460119081722965460, 4834825995463338092669, 153840302781842431823086 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

a(n) = coefficient of x^n in (1+x*A(x))^(5*n+1)/(5*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(5*n+m,k)/(5*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)^4*F(x,n+1)) for n>0 with F(x,0)=1.

EXAMPLE

G.f.: A(x) = 1 + x + 6*x^2 + 56*x^3 + 661*x^4 + 9141*x^5 + 142522*x^6 +...

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

A = 1 + x*A^4*B;

B = A*(1 + x*B^4*C);

C = B*(1 + x*C^4*D);

D = C*(1 + x*D^4*E);

E = D*(1 + x*E^4*F); ...

where B(x) = A(x)*A(x*A(x)^5), C(x) = A(x)*B(x*A(x)^5),  D(x) = A(x)*C(x*A(x)^5), ...

Expansions of a few of the functions described above begin:

B(x) = 1 + 2*x + 18*x^2 + 229*x^3 + 3480*x^4 + 59466*x^5 +...

C(x) = 1 + 3*x + 36*x^2 + 585*x^3 + 11055*x^4 + 230211*x^5 +...

D(x) = 1 + 4*x + 60*x^2 + 1190*x^3 + 27040*x^4 + 669426*x^5 +...

E(x) = 1 + 5*x + 90*x^2 + 2110*x^3 + 56145*x^4 + 1616151*x^5 +...

ALTERNATE GENERATING METHOD.

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

A = 1 + x*A^5*B,

B = 1 + x*(A*B)^5*C,

C = 1 + x*(A*B*C)^5*D,

D = 1 + x*(A*B*C*D)^5*E, etc.,

then B(x) = A(x*A(x)^5), C(x) = B(x*A(x)^5), D(x) = C(x*A(x)^5), etc.,

where A(x) = 1 + x*A(x)^5*A(x*A(x)^5) is the g.f. of this sequence.

Expansions of a few of the functions described above begin:

B(x) = 1 + x + 11*x^2 + 156*x^3 + 2541*x^4 + 45571*x^5 + 881403*x^6 +...

C(x) = 1 + x + 16*x^2 + 306*x^3 + 6446*x^4 + 145201*x^5 +...

D(x) = 1 + x + 21*x^2 + 506*x^3 + 13126*x^4 + 358281*x^5 +...

E(x) = 1 + x + 26*x^2 + 756*x^3 + 23331*x^4 + 750061*x^5 +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^5*subst(A, x, x*A^5+x*O(x^n))); polcoeff(A, n)}

(PARI) /* a(n) = [x^n] (1+x*A(x))^(5*n+1)/(5*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))^(5*m+1)/(5*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(5*n+m, k)/(5*n+m)*a(n-k, k))))}

for(n=0, 41, print1(a(n), ", "))

CROSSREFS

Cf. A088717, A215505, A215506.

Sequence in context: A182955 A053336 A290788 * A112699 A093197 A303921

Adjacent sequences:  A215504 A215505 A215506 * A215508 A215509 A215510

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.)