A192623 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^(2*n))/(1 - x^(n+1)*A(x)^(2*n)).
1, 2, 4, 16, 70, 336, 1720, 9152, 50140, 280882, 1601496, 9263424, 54224312, 320611152, 1912003536, 11487287872, 69463274022, 422440713680, 2582081184572, 15853795192704, 97736576247976, 604744065493936, 3754311394271208, 23377930236777152
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 70*x^4 + 336*x^5 + 1720*x^6 +... The g.f. A = A(x) satisfies the following relations: (0) A = (1+x)/(1-x) * (1+x^2*A^2)/(1-x^2*A^2) * (1+x^3*A^4)/(1-x^3*A^4) * (1+x^4*A^6)/(1-x^4*A^6)*... (1) A = 1 + 2*x/((1-x)*(1-x*A^2)) + 2*x^3*A^2*(1+x*A^2)/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)) + 2*x^6*A^6*(1+x*A^2)*(1+x^2*A^4)/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)*(1-x^3*A^4)*(1-x^3*A^6)) +... (2) A^2 = 1 + 4*x/((1-x)*(1-x*A^2)) + 4*x^2*(1+x*A^2)^2/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)) + 4*x^3*(1+x*A^2)^2*(1+x^2*A^4)^2/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)*(1-x^3*A^4)*(1-x^3*A^6)) +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..230
Programs
-
Mathematica
(* Calculation of constants {d,c}: *) Chop[{1/r, (s*Sqrt[(QPochhammer[r, r*s^2]*(Log[r*s^2] - 2*QPolyGamma[0, Log[-r]/Log[r*s^2], r*s^2] + 2*QPolyGamma[0, Log[r]/Log[r*s^2], r*s^2]))/(Log[ r*s^2]*(QPochhammer[r, r*s^2] - 4*r*s^2*(Derivative[0, 1][QPochhammer][r, r*s^2] + r*s*(-Derivative[0, 2][QPochhammer][-r, r*s^2] + s*Derivative[0, 2][QPochhammer][r, r*s^2]))))])/(2* Sqrt[Pi])} /. FindRoot[{s == QPochhammer[-r, r*s^2]/QPochhammer[r, r*s^2], QPochhammer[r, r*s^2] + 2*r*s^2*Derivative[0, 1][QPochhammer][r, r*s^2] == 2*r*s*Derivative[0, 1][QPochhammer][-r, r*s^2]}, {r, 1/6}, {s, 3/2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *) -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=prod(k=0,n,(1+x^(k+1)*A^(2*k))/(1-x^(k+1)*(A+x*O(x^n))^(2*k))));polcoeff(A,n)} -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^(m*(m+1)/2)*A^(m*(m-1))*prod(k=0,m-1,(1+x^k*A^(2*k))/((1-x^(k+1)*A^(2*k))*(1-x^(k+1)*A^(2*k+2) +x*O(x^n))))));polcoeff(A,n)} -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=sqrt(1+sum(m=1,n,x^m*prod(k=0,m-1,(1+x^k*A^(2*k))^2/((1-x^(k+1)*A^(2*k) +x*O(x^n))*(1-x^(k+1)*A^(2*k+2)))))));polcoeff(A,n)}
Formula
G.f. satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^(n*(n-1)) * Product_{k=0..n-1} (1 + x^k*A(x)^(2*k))/((1 - x^(k+1)*A(x)^(2*k))*(1 - x^(k+1)*A(x)^(2*k+2))), due to the Heine identity.
(2) A(x)^2 = 1 + Sum_{n>=1} x^n * Product_{k=0..n-1} (1 + x^k*A(x)^(2*k))^2/((1 - x^(k+1)*A(x)^(2*k))*(1 - x^(k+1)*A(x)^(2*k+2)), due to the Heine identity.
Self-convolution yields A192622.
a(n) ~ c * d^n / n^(3/2), where d = 6.6513304693895827154026... and c = 0.31031758618989325658... - Vaclav Kotesovec, Jun 30 2025
Comments