A215507 G.f. satisfies: A(x) = 1 + x*A(x)^5*A(x*A(x)^5).
1, 1, 6, 56, 661, 9141, 142522, 2448544, 45653707, 913964706, 19491269046, 440154262428, 10475920613965, 261802864005533, 6848792691398646, 187061918111607286, 5322557388634585229, 157460119081722965460, 4834825995463338092669, 153840302781842431823086
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Keywords
Examples
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 +...
Programs
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Mathematica
terms = 20; A[] = 0; Do[A[x] = 1 + x*A[x]^5 * A[x*A[x]^5] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 31 2025 *)
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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),", "))
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.