cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^n)^2/(1 - x^(n+1)*A(x)^n)^2.

Original entry on oeis.org

1, 4, 12, 48, 220, 1080, 5600, 30112, 166300, 937620, 5374200, 31221488, 183430656, 1087975256, 6505878592, 39179738400, 237412139260, 1446488046824, 8855937880108, 54455375407504, 336159421649528, 2082508824181856, 12942736191473792
Offset: 0

Views

Author

Paul D. Hanna, Jul 06 2011

Keywords

Comments

Related q-series (Heine) identity:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x*A(x), x=x, y=z=1.

Examples

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 48*x^3 + 220*x^4 + 1080*x^5 +...
The g.f. A = A(x) satisfies the following relations:
A = (1+x)^2/(1-x)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A^2)^2/(1-x^3*A^2)^2 *...
A = 1 + 4*x/((1-x)*(1-x*A)) + 4*x^2*(1+x*A)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)) + 4*x^3*(1+x*A)^2*(1+x^2*A^2)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x^3*A^3)) +...

Links

Crossrefs

Cf. A192623, A192620, A192624.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) Chop[{1/r, s*Sqrt[(QPochhammer[r, r*s]*(Log[r*s] - 2*QPolyGamma[0, Log[-r]/Log[r*s], r*s] + 2*QPolyGamma[0, Log[r]/Log[r*s], r*s]))/(Pi* Log[r*s]*(QPochhammer[r, r*s] - 4*r*s*(Derivative[0, 1][QPochhammer][r, r*s] - r*Sqrt[s]*Derivative[0, 2][QPochhammer][-r, r*s] + r*s*Derivative[0, 2][QPochhammer][r, r*s])))]} /. FindRoot[{QPochhammer[-r, r*s]^2/QPochhammer[r, r*s]^2 == s, QPochhammer[r, r*s]^2 + 2*r*s*QPochhammer[r, r*s] * Derivative[0, 1][QPochhammer][r, r*s] == 2*r*QPochhammer[-r, r*s] * Derivative[0, 1][QPochhammer][-r, r*s]}, {r, 1/6}, {s, 3}, 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^k)^2/(1-x^(k+1)*(A+x*O(x^n))^k)^2));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*prod(k=0,m-1,(1+x^k*A^k)^2/((1-x^(k+1)*A^k +x*O(x^n))*(1-x^(k+1)*A^(k+1))))));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (1 + x^k*A(x)^k)^2/((1 - x^(k+1)*A(x)^k)*(1 - x^(k+1)*A(x)^(k+1)) due to the Heine identity.
Self-convolution of A192623.
a(n) ~ c * d^n / n^(3/2), where d = 6.65133046938958271... and c = 1.095759838870545... - Vaclav Kotesovec, Jun 30 2025

A192620 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/(1 - x^n*A(x))^2.

Original entry on oeis.org

1, 4, 28, 224, 1948, 17928, 171776, 1695872, 17133436, 176297668, 1841222776, 19467629120, 207978652416, 2241618514120, 24345336854400, 266168049520832, 2927074607294300, 32356419163487336, 359330087240388828, 4007079691584624576
Offset: 0

Views

Author

Paul D. Hanna, Jul 06 2011

Keywords

Comments

Related q-series (Heine) identity:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x, x=x*A(x), y=z=1.

Examples

			G.f.: A(x) = 1 + 4*x + 28*x^2 + 224*x^3 + 1948*x^4 + 17928*x^5 + ...
The g.f. A = A(x) satisfies the following relations:
(0) A = (1+x*A)^2/(1-x*A)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A)^2/(1-x^3*A)^2 * ...
(1) A = 1 + 4*x*A/((1-x*A)*(1-x)) + 4*x^2*A^2*(1+x)^2/((1-x*A)*(1-x^2*A)*(1-x)*(1-x^2)) + 4*x^3*A^3*(1+x)^2*(1+x^2)^2/((1-x*A)*(1-x^2*A)*(1-x^3*A)*(1-x)*(1-x^2)*(1-x^3)) + ...
(2) A^(1/2) = 1 + 2*x*A/(1-x) + 2*x^2*A^2*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^3*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) + ...

Links

Crossrefs

Cf. A190862, A192621, A192622, A192623, A192624, A192625.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x])^2/(1 - x^k*A[x])^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 04 2023 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(r*s^(3/2)*((-1 + s)*Derivative[0, 1][QPochhammer][-s, r] + Sqrt[s]*(1 + s)*Derivative[0, 1][QPochhammer][s, r]))/(2* Pi*(1 + s)*QPochhammer[s, r]* (2* s*((1 + s^2)/(-1 + s^2)^2) + (QPolyGamma[1, Log[-s]/Log[r], r] - QPolyGamma[1, Log[s]/Log[r], r])/ Log[r]^2))]} /. FindRoot[{(-1 + s)^2*(QPochhammer[-s, r]^2/((1 + s)^2*QPochhammer[s, r]^2)) == s, 1 - 4*(s/(-1 + s^2)) + (2*(QPolyGamma[0, Log[-s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]))/Log[r] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Mar 03 2024 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=prod(k=1,n,(1+x^k*A)^2/(1-x^k*A+x*O(x^n))^2));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^m*prod(k=1,m,(1+x^(k-1))^2/((1-x^k*A+x*O(x^n))*(1-x^k)))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1+sum(m=1,n,x^m*A^m*prod(k=1,m,(1+x^(k-1))/(1-x^k+x*O(x^n)))))^2);polcoeff(A,n)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x))*(1-x^k), due to the Heine identity.
(2) A(x)^(1/2) = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.
Equals the self-convolution of A192621.
a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 1.04958502741924123967536156787764354342367951743839... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 2.638555772492608872250287025192536127217... satisfy A(r) = 1 / Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^2) and A(r) = Product_{n>=1} (1 + r^n*A(r))^2/(1 - r^n*A(r))^2. - Paul D. Hanna, Mar 02 2024

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)).

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Jul 06 2011

Keywords

Comments

Related q-series (Heine) identity:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x*A(x)^2, x=x, y=1, z=0.

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

Crossrefs

Cf. A192622 (g.f. A(x)^2), A192621, A192624.

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

A192625 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/((1-x^n)*(1 - x^n*A(x)^2)).

Original entry on oeis.org

1, 4, 28, 240, 2348, 24952, 280192, 3271232, 39310668, 483032980, 6041149272, 76648727632, 984161689728, 12764078032568, 166969699620640, 2200415358484800, 29186416580736300, 389340777798701672, 5220028320540100220, 70303231772070200912
Offset: 0

Views

Author

Paul D. Hanna, Jul 06 2011

Keywords

Comments

Related q-series (Heine) identity:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)),
here q=x, x=x, y=z=A(x).

Examples

			G.f.: A(x) = 1 + 4*x + 28*x^2 + 240*x^3 + 2348*x^4 + 24952*x^5 +...
The g.f. A = A(x) satisfies:
A = (1+x*A)^2/((1-x)*(1-x*A^2)) * (1+x^2*A)^2/((1-x^2)*(1-x^2*A^2)) * (1+x^3*A)^2/((1-x^3)*(1-x^3*A^2)) *...
A = {1 + x*(A+1)^2/(1-x)^2 + x^2*(A+1)^2*(A+x)^2/((1-x)*(1-x^2))^2 + x^3*(A+1)^2*(A+x)^2*(A+x^2)^2/((1-x)*(1-x^2)*(1-x^3))^2 +...

Links

Crossrefs

Cf. A192624, A192620, A192622.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) Chop[{1/r, (1/(2*Sqrt[Pi]))*s*(-1 + s^2)* Sqrt[(Log[ r]*(r*Log[r]*QPochhammer[-s, r]*QPochhammer[s^2, r] * Derivative[0, 1][QPochhammer][r, r] + QPochhammer[r, r]*(-2*r*Log[r]*QPochhammer[s^2, r]* Derivative[0, 1][QPochhammer][-s, r] + QPochhammer[-s, r]*((-QPochhammer[s^2, r])*(Log[1 - r] + QPolyGamma[0, 1, r]) + r*Log[r] * Derivative[0, 1][QPochhammer][s^2, r]))))/(QPochhammer[ r, r]*QPochhammer[-s, r]*QPochhammer[s^2, r] * (s*(1 + s^2) * Log[r]^2 + (-1 + s^2)^2 * QPolyGamma[1, Log[-s]/Log[r], r] - 2*(-1 + s^2)^2 * QPolyGamma[1, (2*Log[s])/Log[r], r]))]} /. FindRoot[{s + ((-1 + s) * QPochhammer[-s, r]^2)/((1 + s) * QPochhammer[r, r] * QPochhammer[s^2, r]) == 0, (2*s)/(-1 + s^2) + (2*(-QPolyGamma[0, Log[-s]/Log[r], r] + QPolyGamma[0, (2*Log[s])/Log[r], r]))/Log[r] == 1}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=prod(k=1,n,(1+x^k*A)^2/((1-x^k+x*O(x^n))*(1-x^k*A^2))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*prod(k=0,m-1,(A+x^k)^2/(1-x^(k+1)+x*O(x^n))^2)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (A(x) + x^k)^2/(1-x^(k+1))^2 due to the Heine identity.
a(n) ~ c * d^n / n^(3/2), where d = 14.589835921559349866989401284706614286378779559... and c = 0.453836352355009937995115064134624562210185... - Vaclav Kotesovec, Jun 30 2025
Showing 1-4 of 4 results.

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