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 1-3 of 3 results.

A384272 G.f. A(x) satisfies -2*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

Original entry on oeis.org

1, 2, 2, 6, 16, 50, 144, 478, 1510, 5116, 17034, 58812, 202166, 709228, 2489546, 8848146, 31525526, 113236920, 407983964, 1478249454, 5372468156, 19607233026, 71758722172, 263480958508, 969856453650, 3579426292768, 13239549874552, 49078409375334, 182282423994240, 678289439131812, 2528257204808848
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2025

Keywords

Comments

The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 16*x^4 + 50*x^5 + 144*x^6 + 478*x^7 + 1510*x^8 + 5116*x^9 + 17034*x^10 + 58812*x^11 + 202166*x^12 + ...
where
-2*x = (1 - x/A(x))*(1 - A(x))*(1+x) * (1 - x^2/A(x))*(1 - x*A(x))*(1+x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1+x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1+x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1+x^5) * ...

Links

Crossrefs

Cf. A356499, A384271, A384273.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) With[{k = 2}, Chop[{1/r, (1/Sqrt[2*Pi])*(-1 + s)* Sqrt[(s^2*(-r + s)*Log[r]*((r - s)*Log[1 - r] - r*Log[r] + (r - s)*(QPolyGamma[0, -1 + Log[s]/Log[r], r] + r*Log[r]*(Derivative[0, 1][QPochhammer][-1, r]/ QPochhammer[-1, r] + Derivative[0, 1][QPochhammer][1/s, r]/ QPochhammer[1/s, r] + Derivative[0, 1][QPochhammer][s/r, r]/ QPochhammer[s/r, r])))) / (-2*s*(r + r^2 - 3*r*s + s^3)* Log[r]^2 + 2*(-1 + s)*(-r + s)*(-r + s^2)* Log[r]*(QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) + (r - s)^2*(-1 + s)^2*((QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) * (Log[r] - QPolyGamma[0, -Log[s]/Log[r], r] + QPolyGamma[0, -1 + Log[s]/Log[r], r]) + QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, -1 + Log[s]/Log[r], r]))]} /. FindRoot[{2*k* r + (r*s*QPochhammer[-1, r]*QPochhammer[1/s, r]* QPochhammer[s/r, r])/((r - s)*(-1 + s)) == 0, (-r + s^2)*Log[r] + (r - s)*(-1 + s) * QPolyGamma[0, Log[1/s]/Log[r], r] - (r - s)*(-1 + s)*QPolyGamma[0, Log[s/r]/Log[r], r] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120]]] (* Vaclav Kotesovec, Jun 30 2025 *)
  • PARI
    {a(n) = my(A=[1,2]);  for(i=2,n, A=concat(A,0);
A[#A] = polcoef(2*x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.
(1) -2*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
(2) 2*x/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
(3) -2*x*theta_4(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).
(4) 2*x*theta_4(x)/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).
(5.a) -2*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(5.b) -2*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(6.a) 2*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.
(6.b) 2*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
a(n) ~ c * d^n / n^(3/2), where d = 3.9182818074503417233561248171647191927022193746074095378101... and c = 0.687770752477136312107316168146720576083024421405682875987... - Vaclav Kotesovec, Jun 30 2025

A384273 G.f. A(x) satisfies -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

Original entry on oeis.org

1, 3, 3, 9, 39, 108, 387, 1581, 5196, 21573, 82596, 318279, 1303146, 5182389, 20919156, 86577264, 351929133, 1462075095, 6077250693, 25277372124, 106131459906, 445859648019, 1878449392365, 7955646845046, 33707865532680, 143344958486019, 610977896794104, 2608218534504888, 11162376089875158
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2025

Keywords

Comments

The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

Examples

			G.f.: A(x) = 1 + 3*x + 3*x^2 + 9*x^3 + 39*x^4 + 108*x^5 + 387*x^6 + 1581*x^7 + 5196*x^8 + 21573*x^9 + 82596*x^10 + 318279*x^11 + 1303146*x^12 + ...
where
-3*x = (1 - x/A(x))*(1 - A(x))*(1+x) * (1 - x^2/A(x))*(1 - x*A(x))*(1+x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1+x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1+x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1+x^5) * ...

Links

Crossrefs

Cf. A356499, A384271, A384272.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) With[{k = 3}, Chop[{1/r, (1/Sqrt[2*Pi])*(-1 + s)* Sqrt[(s^2*(-r + s)*Log[r]*((r - s)*Log[1 - r] - r*Log[r] + (r - s)*(QPolyGamma[0, -1 + Log[s]/Log[r], r] + r*Log[r]*(Derivative[0, 1][QPochhammer][-1, r]/ QPochhammer[-1, r] + Derivative[0, 1][QPochhammer][1/s, r]/ QPochhammer[1/s, r] + Derivative[0, 1][QPochhammer][s/r, r]/ QPochhammer[s/r, r])))) / (-2*s*(r + r^2 - 3*r*s + s^3) * Log[r]^2 + 2*(-1 + s)*(-r + s)*(-r + s^2)* Log[r]*(QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) + (r - s)^2*(-1 + s)^2*((QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) * (Log[r] - QPolyGamma[0, -Log[s]/Log[r], r] + QPolyGamma[0, -1 + Log[s]/Log[r], r]) + QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, -1 + Log[s]/Log[r], r]))]} /. FindRoot[{2*k* r + (r*s*QPochhammer[-1, r]*QPochhammer[1/s, r] * QPochhammer[s/r, r])/((r - s)*(-1 + s)) == 0, (-r + s^2)*Log[r] + (r - s)*(-1 + s) * QPolyGamma[0, Log[1/s]/Log[r], r] - (r - s)*(-1 + s)*QPolyGamma[0, Log[s/r]/Log[r], r] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120]]] (* Vaclav Kotesovec, Jun 30 2025 *)
  • PARI
    {a(n) = my(A=[1,3]);  for(i=2,n, A=concat(A,0);
A[#A] = polcoef(3*x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.
(1) -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
(2) 3*x/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
(3) -3*x*theta_4(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).
(4) 3*x*theta_4(x)/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).
(5.a) -3*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(5.b) -3*x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(6.a) 3*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.
(6.b) 3*x*theta_4(x)/A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.51389712142676799318649101918382471452597633164586338354... and c = 0.80137914461001211839391539872328025866773400116571811811... - Vaclav Kotesovec, Jun 30 2025

A384269 G.f. A(x) satisfies x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).

Original entry on oeis.org

1, 1, 2, 6, 16, 49, 154, 513, 1747, 6078, 21439, 76607, 276685, 1008781, 3707512, 13721086, 51088860, 191245836, 719333008, 2717229481, 10303797518, 39208957744, 149676496756, 573037914270, 2199735075908, 8464921506665, 32648239747059, 126185248269567, 488657718553676, 1895790377527674
Offset: 0

Views

Author

Paul D. Hanna, May 25 2025

Keywords

Comments

The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 49*x^5 + 154*x^6 + 513*x^7 + 1747*x^8 + 6078*x^9 + 21439*x^10 + ...
RELATED SERIES.
1/A(x) = 1 - x - x^2 - 3*x^3 - 5*x^4 - 16*x^5 - 45*x^6 - 155*x^7 - 512*x^8 - 1763*x^9 - 6084*x^10 + ...
SPECIFIC VALUES.
A(exp(-Pi)) = 1.0474973549949421045732567080496722542518531011526934631...
  where Sum_{n=-oo..+oo} (-1)^n * exp(-Pi*n*(n+1)/2) * A(exp(-Pi))^n = exp(-Pi) * (Pi/2)^(1/4) / gamma(3/4) = 0.03947933420376592813...
A(-exp(-Pi)) = 0.960086060200580366759936974556134222228793624085744940...
  where Sum_{n=-oo..+oo} (-1)^(n*(n-1)/2) * exp(-Pi*n*(n+1)/2) * A(-exp(-Pi))^n = -exp(-Pi) * Pi^(1/4) / gamma(3/4) = -0.04694910513068872743...
A(t) = 2 at t = 0.24484187571695922418922496399796775078115821427621282...
A(t) = 7/4 at t = 0.239324355731620083092236573970947000576283799760943...
A(t) = 5/3 at t = 0.234439889083627870257298020352799276294012688627782...
A(t) = 3/2 at t = 0.217134571709901433113197085617818478214816713922905...
A(t) = 4/3 at t = 0.183806911401666173138177455971709388630788740531594...
A(t) = 5/4 at t = 0.157416870441717618165825450612923233287765184975643...
A(1/5) = 1.401449039483961854381757985869052435618161722574956...
A(1/6) = 1.276318946972284528693666572724710434062725174240448...
A(1/7) = 1.213287805382388838362413216213677242108560133326140...
A(1/8) = 1.174388177498186580244775740286834758637341200438483...
A(1/9) = 1.147764942051942680447238410304951699474657455354304...

Links

Crossrefs

Cf. A356499, A384271.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, -s*Log[r]/2* Sqrt[((s-1)*(-2*r*(s - 1) * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][-1, r] + QPochhammer[-1, r]^2 * QPochhammer[s, r]^2 * Derivative[0, 1][QPochhammer][1/s, r] + 2*(s-1)* QPochhammer[-1, r] * (QPochhammer[s, r] - r*Derivative[0, 1][QPochhammer][s, r])))/ (Pi*QPochhammer[-1, r] * QPochhammer[s, r] * (-s*Log[ r]^2 + (s-1)^2 * (QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, Log[s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1, r] * QPochhammer[1/s, r] * QPochhammer[s, r] == 2*r*(1 - s), s*Log[r] + (s-1) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/s]/Log[r], r]) == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jun 01 2025 *)
  • PARI
    {a(n) = my(A=[1,1]);  for(i=2,n, A=concat(A,0);
A[#A] = polcoef(x - prod(n=1,#A, (1 - x^n*Ser(A)) * (1 - x^(n-1)/Ser(A)) * (1 + x^n) ),#A-1); ); H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.
(1) x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
(2) -x*A(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
(3) x*theta_4(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).
(4) -x*theta_4(x)*A(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).
(5.a) x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(5.b) x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.
(6.a) -x*theta_4(x)*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(6.b) -x*theta_4(x)*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.083846421711383847604673417919116998017... and c = 0.584432537831593677040363592052688856... - Vaclav Kotesovec, Jun 01 2025
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