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 1-6 of 6 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

A192621 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, 2, 12, 88, 726, 6456, 60392, 585792, 5838764, 59440250, 615431464, 6460681656, 68607630680, 735682014648, 7954732578032, 86635206695808, 949518438959574, 10464751843723840, 115904823140622164, 1289419736206548408, 14401729960605163272
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)^2, y=1, z=0.

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 88*x^3 + 726*x^4 + 6456*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 + 2*x*A^2/(1-x) + 2*x^2*A^4*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^6*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) + ...
(2) A = 1 + 2*x*A^2/((1-x*A^2)*(1-x)) + 2*x^3*A^4*(1+x)/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 2*x^6*A^6*(1+x)*(1+x^2)/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...
(3) A^2 = 1 + 4*x*A^2/((1-x*A^2)*(1-x)) + 4*x^2*A^4*(1+x)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 4*x^3*A^6*(1+x)^2*(1+x^2)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...

Links

Crossrefs

Cf. A192620 (g.f. A(x)^2), A192623, A190862, A196150, A196151.

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 constant d: *) 1/r /. FindRoot[{QPochhammer[-s^2, r] / QPochhammer[s^2, r] == s*((1 + s^2)/(1 - s^2)), QPolyGamma[0, Log[-s^2]/Log[r], r] - QPolyGamma[0, Log[s^2]/Log[r], r] == (2*(s^2/(s^4 - 1)) - 1/2) * Log[r]}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)
  • 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^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*A^(2*m)*prod(k=1,m,(1+x^(k-1))^2/((1-x^k*A^2 +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*(m+1)/2)*A^(2*m)*prod(k=1,m,(1+x^(k-1))/((1-x^k*A^2 +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^(2*m)*prod(k=1,m,(1+x^(k-1))/(1-x^k+x*O(x^n)))) );polcoeff(A,n)}

Formula

G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.
(2) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/((1-x^k*A(x)^2)*(1-x^k)), due to the Heine identity.
(3) A(x)^2 = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x)^2)*(1-x^k), due to the Heine identity.
Self-convolution yields A192620.
a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 0.323075847195701225672585138139173170517867693... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 1.624363189835514855585723923742556266289... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^4) ) and A(r) = Product_{n>=1} (1 + r^n*A(r)^2)/(1 - r^n*A(r)^2). - Paul D. Hanna, Mar 02 2024

A302171 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x))^k.

Original entry on oeis.org

1, 1, 4, 14, 54, 213, 880, 3724, 16143, 71227, 319067, 1447160, 6633530, 30682425, 143028870, 671293632, 3169572659, 15044993968, 71752624923, 343658572717, 1652266087698, 7971518032791, 38581202763318, 187269381724629, 911404238805468, 4446493502832481, 21742327471261176
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 02 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 213*x^5 + 880*x^6 + 3724*x^7 + 16143*x^8 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x))^2 * (1 - x^3*A(x))^3 * ...).

Links

Crossrefs

Cf. A000219, A145267, A145268, A190862, A298260, A298261, A301455, A301624, A301831.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1/Product[(1 - x^k*A[x])^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 5.177446537296361283814259811908762546749... and c = 0.81395777803098291048009263980507199... - Vaclav Kotesovec, Sep 27 2023
Radius of convergence r = 0.1931454033945844258723936803941781838... = 1/d and A(r) = 2.2252305561396523944672847657756264073... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 - r^n*A(r)) and (2) A(r) = 1 / Product_{n>=1} (1 - r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024

A190861 G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x))/(1-x^n).

Original entry on oeis.org

1, 2, 6, 18, 56, 178, 580, 1922, 6466, 22022, 75788, 263152, 920768, 3243414, 11492460, 40934616, 146484296, 526389182, 1898722242, 6872300848, 24951521464, 90851221740, 331666951116, 1213729811070, 4451547793956
Offset: 0

Views

Author

Paul D. Hanna, May 21 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 178*x^5 + 580*x^6 + ...
such that the g.f. satisfies the identity:
A(x) = (1+x*A(x))/(1-x) * (1+x^2*A(x))/(1-x^2) * (1+x^3*A(x))/(1-x^3) * ...
A(x) = 1 + x*(1+A(x))/(1-x) + x^2*(1+A(x))*(1+x*A(x))/((1-x)*(1-x^2)) + x^3*(1+A(x))*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)*(1-x^3)) + ...

Crossrefs

Cf. A209357, A190862.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x])/(1-x^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 04 2023 *)
(* Calculation of constants {d,c}: *) {1/r, Sqrt[-(s*(1 + s)* Log[r]*(s*(1 + s)*QPochhammer[r]^2*(Log[1 - r] + QPolyGamma[0, 1, r]) - r*Log[r]*QPochhammer[-s, r] * Derivative[0, 1][QPochhammer][r, r] + r*Log[r]*QPochhammer[r] * Derivative[0, 1][QPochhammer][-s, r]))/(2*Pi * QPochhammer[r]^2 * (s*Log[r]^2 + (1 + s)^2 * QPolyGamma[1, Log[-s]/Log[r], r]))]} /. FindRoot[{s*(1 + s)*QPochhammer[r] == QPochhammer[-s, r], Log[1 - r] + (1 + 2*s)*Log[r]/(1 + s) + QPolyGamma[0, Log[-s]/Log[r], r] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = prod(m=1,n, (1 + x^m*A)/(1 - x^m +x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = 1 + sum(m=1,n, x^m*prod(k=1,m, (1 + x^(k-1)*A)/(1 - x^k +x*O(x^n))))); polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=1..n} (1 + x^(k-1)*A(x))/(1-x^k) due to the q-binomial theorem.
a(n) ~ c * d^n / n^(3/2), where d = 3.881937422067584825536867239508405299121... and c = 4.5308041082663146457769... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.25760332825442180041464062514057254352... and A(r) = 5.79064730997128469298918813333150154669... satisfy A(r) = 1 / Sum_{n>=1} r^n/(1 + r^n*A(r)) and A(r) = Product_{n>=1} (1 + r^n*A(r))/(1-r^n). Note that r = 1/d and A(r) = s as given in the Mathematica program by Vaclav Kotesovec. - Paul D. Hanna, Mar 04 2024

A192619 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^(1/2))^2/(1 - x^n*A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 20, 104, 556, 3048, 17064, 97216, 562036, 3289836, 19461448, 116178600, 699045176, 4235292680, 25816944176, 158223753376, 974389668364, 6026623271840, 37420762694588, 233179517592232, 1457706542138344, 9139698522931008
Offset: 0

Views

Author

Paul D. Hanna, Jul 06 2011

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 20*x^2 + 104*x^3 + 556*x^4 + 3048*x^5 +...
The g.f. A = A(x) satisfies:
A = (1+x*A^(1/2))^2/(1-x*A^(1/2))^2 * (1+x^2*A^(1/2))^2/(1-x^2*A^(1/2))^2 * (1+x^3*A^(1/2))^2/(1-x^3*A^(1/2))^2 *...

Links

Crossrefs

Cf. A190862.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) Chop[{1/r, (1 - s)*s*Log[r]* Sqrt[(2*r*(QPochhammer[Sqrt[s], r]* Derivative[0, 1][QPochhammer][-Sqrt[s], r] - QPochhammer[-Sqrt[s], r]* Derivative[0, 1][QPochhammer][Sqrt[s], r]))/ (Pi* QPochhammer[s, r^2]*(-2*Sqrt[s]*(1 + s)*Log[r]^2 + (-1 + s)^2* QPolyGamma[1, Log[s]/Log[r^2], r] - (-1 + s)^2* QPolyGamma[1, (2*I*Pi + Log[s])/Log[r^2], r]))]} /. FindRoot[{((-1 + Sqrt[s])^2* QPochhammer[-Sqrt[s], r]^2)/((1 + Sqrt[s])^2* QPochhammer[Sqrt[s], r]^2) == s, (-1 - 2*Sqrt[s] + s)/(-1 + s) + (QPolyGamma[0, Log[-Sqrt[s]]/Log[r], r] - QPolyGamma[0, Log[s]/(2*Log[r]), r])/ Log[r] == 0}, {r, 1/6}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+x^m*A^(1/2))^2/(1-x^m*A^(1/2)+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/2)*prod(k=1, m,(1+x^(k-1))/(1-x^k+x*O(x^n)))))^2); polcoeff(A, n)}

Formula

Self-convolution of A190862.
a(n) ~ c * d^n / n^(3/2), where d = 6.693428901114353533300254329706934045134... and c = 4.7342954578062245798237099751798009... - Vaclav Kotesovec, Jun 30 2025

A209357 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^(n+1)*A(x)) / (1 - x^n).

Original entry on oeis.org

1, 1, 3, 6, 14, 31, 72, 166, 390, 922, 2197, 5273, 12728, 30892, 75327, 184476, 453505, 1118798, 2768843, 6872437, 17103411, 42670102, 106697009, 267359854, 671260241, 1688411587, 4254084396, 10735614274, 27132998096, 68671994940, 174035109012, 441607820562
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 72*x^6 + 166*x^7 +...
where the g.f. satisfies the identity:
A(x) = (1+x^2*A(x))/(1-x) * (1+x^3*A(x))/(1-x^2) * (1+x^4*A(x))/(1-x^3) *...
A(x) = 1 + x*(1+x*A(x))/(1-x) + x^2*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^3*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) +...

Links

Crossrefs

Cf. A190861, A190862.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(s*(1 + r*s)*Log[r]*(s*(1 + r*s)*(-QPochhammer[r]*(Log[1 - r] + Log[r] + QPolyGamma[0, 1, r]) + r*Log[r]*Derivative[0, 1][QPochhammer][r, r]) - r*Log[r]*Derivative[0, 1][QPochhammer][-r*s, r])) / (2*Pi*QPochhammer[r] * (r*s*Log[r]^2 + (1 + r*s)^2*QPolyGamma[1, Log[-r*s]/Log[r], r]))]} /. FindRoot[{s*(1 + r*s) == QPochhammer[-r*s, r]/QPochhammer[r], Log[1-r] + r*s*Log[r]/(1 + r*s) + QPolyGamma[0, Log[-r*s]/Log[r], r] == -Log[r]}, {r, 2/5}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 10 2025 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+x^(m+1)*A)/(1-x^m+x*O(x^n)))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*prod(k=1, m, (1+x^k*A)/(1-x^k+x*O(x^n))))); polcoeff(A, n)}
for(n=0,35,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^n*Product_{k=1..n} (1 + x^k*A(x))/(1-x^k) due to the q-binomial theorem.
a(n) ~ c * d^n / n^(3/2), where d = 2.6481816651621274063587047915... and c = 7.257947883786923940523402074... - Vaclav Kotesovec, Jun 10 2025
Showing 1-6 of 6 results.

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