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 1-5 of 5 results.

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

A192624 G.f. satisfies: A(x) = Product_{n>=1} (1+x^n)*(1 + x^n*A(x))/((1-x^n)*(1 - x^n*A(x))).

Original entry on oeis.org

1, 4, 20, 112, 676, 4328, 28912, 199392, 1409364, 10157828, 74375640, 551715264, 4137527408, 31318286632, 238958947328, 1835960454272, 14192132860868, 110298595778872, 861338925309604, 6755283201399776, 53185599585579640
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=A(x), z=1.

Examples

			G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4328*x^5 +...
The g.f. A = A(x) satisfies:
A = (1+x)*(1+x*A)/((1-x)*(1-x*A)) * (1+x^2)*(1+x^2*A)/((1-x^2)*(1-x^2*A)) * (1+x^3)*(1+x^3*A)/((1-x^3)*(1-x^3*A)) *...
A = {1 + 2*x*(A+1)/(1-x)^2 + 2*x^2*(A+1)*(A+x)*(1+x)/((1-x)*(1-x^2))^2 + 2*x^3*(A+1)*(A+x)*(A+x^2)*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3))^2 +...

Links

Crossrefs

Cf. A192625, A192620, A192622.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) Chop[{1/r, (1/(2*Sqrt[Pi]))* Sqrt[((-1 + s)^2*s*(1 + s)* Log[r]*(-2*r*s*(1 + s)*Log[r]*QPochhammer[r, r]*QPochhammer[s, r]* Derivative[0, 1][QPochhammer][-1, r] + r*(-1 + s)*Log[r]*QPochhammer[-1, r]^2* Derivative[0, 1][QPochhammer][-s, r] + 2*s*(1 + s)*QPochhammer[-1, r]* (r*Log[r]*QPochhammer[s, r]* Derivative[0, 1][QPochhammer][r, r] + QPochhammer[r, r]* ((-QPochhammer[s, r])*(Log[1 - r] + QPolyGamma[0, 1, r]) + r*Log[r]*Derivative[0, 1][QPochhammer][s, r]))))/ (QPochhammer[-1, r]*QPochhammer[r, r]* QPochhammer[s, r]*(2*s*(1 + s^2)*Log[r]^2 + (-1 + s^2)^2* QPolyGamma[1, Log[-s]/Log[r], r] - (-1 + s^2)^2* QPolyGamma[1, Log[s]/Log[r], r]))]} /. FindRoot[{2* s + ((-1 + s)*QPochhammer[-1, r]*QPochhammer[-s, r])/((1 + s)* QPochhammer[r, r]*QPochhammer[s, r]) == 0, (2*s)/(-1 + s^2) + (-QPolyGamma[0, Log[-s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r])/Log[r] == 1}, {r, 1/8}, {s, 3}, 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)*(1+x^k)/((1-x^k+x*O(x^n))*(1-x^k*A))));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)*(1+x^k)/(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)*(1+x^k)/(1-x^(k+1))^2 due to the Heine identity.
a(n) ~ c * d^n / n^(3/2), where d = 8.48061441094248725740275431395... and c = 1.3507361818605928928927572... - 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

A192626 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)^2/(1 - x*(x+x^2)^n)^2.

Original entry on oeis.org

1, 4, 12, 36, 100, 264, 676, 1684, 4096, 9764, 22888, 52872, 120540, 271600, 605556, 1337320, 2927720, 6358432, 13707916, 29351536, 62450468, 132090356, 277845120, 581405140, 1210688864, 2509483020, 5178969644, 10644112012, 21790816340, 44444609044
Offset: 0

Views

Author

Paul D. Hanna, Jul 16 2011

Keywords

Comments

Related q-series identity due to Heine:
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^2, x=x, y=z=1.

Examples

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 36*x^3 + 100*x^4 + 264*x^5 + 676*x^6 +...
where the g.f. equals the product:
A(x) = (1+x)^2/(1-x)^2 * (1+x^2*(1+x))^2/(1-x^2*(1+x))^2 * (1+x^3*(1+x)^2)^2/(1-x^3*(1+x)^2)^2 * (1+x^4*(1+x)^3)^2/(1-x^4*(1+x)^3)^2 *...
which is also equal to the sum:
A(x) = 1 + 4*x/((1-x)*(1-x*(1+x))) + 4*x^2*(1+x*(1+x))^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)) + 4*x^3*(1+x*(1+x))^2*(1 + x^2*(1+x)^2)^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)*(1-x^3*(1+x)^2)*(1-x^3*(1+x)^3)) +...

Crossrefs

Cf. A192627, A192622.

Programs

  • PARI
    {a(n)=local(A=1+x);A=prod(k=0,n+1,(1+x*(x+x^2)^k)^2/(1-x*(x+x^2+x*O(x^n))^k)^2);polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);A=1+sum(m=1,n,x^m*prod(k=0,m-1,(1+(x+x^2)^k)^2/((1-x*(x+x^2)^k +x*O(x^n))*(1-(x+x^2)^(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*(1+x)^k)^2/((1 - x^(k+1)*(1+x)^k)*(1 - x^(k+1)*(1+x)^(k+1))) due to the Heine identity.
Self-convolution of A192627.
Showing 1-5 of 5 results.

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