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.

A268299 G.f. A(x) satisfies: -1 = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n/x) * (1 - A(x)^(n-1)*x).

Original entry on oeis.org

2, 7, 84, 1240, 20942, 382344, 7354688, 146810440, 3012778758, 63167322872, 1347251937632, 29138746861200, 637584335364362, 14088532800477752, 313936020646727040, 7046500093908958288, 159171390375064583380, 3615669944253537267048, 82541551931101193203004, 1892725670848222011475776, 43575217427267416453289838, 1006843304895182755611475824, 23340548167572913996786290328
Offset: 1

Views

Author

Paul D. Hanna, Feb 26 2016

Keywords

Comments

The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.
Equals the row sums of triangle A354650. - Paul D. Hanna, Jul 27 2022

Examples

			G.f.: A(x) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...
where A(x) satisfies the Jacobi Triple Product:
-1 = (1-A(x))*(1-A(x)/x)*(1-x) * (1-A(x)^2)*(1-A(x)^2/x)*(1-A(x)*x) * (1-A(x)^3)*(1-A(x)^3/x)*(1-A(x)^2*x) * (1-A(x)^4)*(1-A(x)^4/x)*(1-A(x)^3*x) * (1-A(x)^5)*(1-A(x)^5/x)*(1-A(x)^4*x) * (1-A(x)^6)*(1-A(x)^6/x)*(1-A(x)^5*x) +...
also
1/x = (1-A(x))*(1-A(x)*x)*(1-1/x) * (1-A(x)^2)*(1-A(x)^2*x)*(1-A(x)/x) * (1-A(x)^3)*(1-A(x)^3*x)*(1-A(x)^2/x) * (1-A(x)^4)*(1-A(x)^4*x)*(1-A(x)^3/x) * (1-A(x)^5)*(1-A(x)^5*x)*(1-A(x)^4/x) * (1-A(x)^6)*(1-A(x)^6*x)*(1-A(x)^5/x) *...
further,
-1 = (1-x) - A(x)*(1-x^3)/x + A(x)^3*(1-x^5)/x^2 - A(x)^6*(1-x^7)/x^3 + A(x)^10*(1-x^9)/x^4 - A(x)^15*(1-x^11)/x^5 + A(x)^21*(1-x^13)/x^6 +...
RELATED SERIES.
The series reversion of g.f. A(x) equals x*Q(x), where Q(x) begins:
Q(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 - 1979248977748/2*x^10/4^10 +...+ A268301(n)/2*x^n/4^n +...
and where Q(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x*Q(x))*(1-1/Q(x)) * (1-x^2)*(1-x^2*Q(x))*(1-x/Q(x)) * (1-x^3)*(1-x^3*Q(x))*(1-x^2/Q(x)) * (1-x^4)*(1-x^4*Q(x))*(1-x^3/Q(x)) * (1-x^5)*(1-x^5*Q(x))*(1-x^4/Q(x)) * (1-x^6)*(1-x^6*Q(x))*(1-x^5/Q(x)) *...

Links

Crossrefs

Cf. A354650, A268300, A268301, A268650.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{r*QPochhammer[1/r, s]*QPochhammer[r, s]* QPochhammer[s, s] == 1 - r, (Log[1 - s] + QPolyGamma[0, 1, s])/(s*Log[s]) - Derivative[0, 1][QPochhammer][1/r, s]/QPochhammer[1/r, s] - Derivative[0, 1][QPochhammer][r, s]/QPochhammer[r, s] - Derivative[0, 1][QPochhammer][s, s]/ QPochhammer[s, s] == 0}, {r, 1/24}, {s, 1/8}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
  • PARI
    {a(n) = my(Q=1/2, t=floor(sqrt(2*n+1)+1/2)); for(i=0, n, Q = (Q + sum(m=-t, t, x^(m*(m-1)/2) * (-Q)^m +x*O(x^n)) )/2 ); polcoeff(serreverse(x*Q), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) -1 = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-x)^n.
(2) -1 = Sum_{n>=0} A(x)^(n*(n+1)/2) * (1 - x^(2*n+1)) / (-x)^n.
(3) 1/x = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-1/x)^n.
(4) 1/x = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n*x) * (1 - A(x)^(n-1)/x).
(5) A(x) = Series_Reversion( x*Q(x) ), where Q(x) is the g.f. of A268301 and satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n*Q(x)) * (1 - x^(n-1)/Q(x)).
(6) x = Sum_{n>=1} a(n) * x^n * Q(x)^n, where Q(x) = Sum_{n>=0} A268301(n)/2*(x/4)^n.
a(n) ~ c * d^n / n^(3/2), where d = 24.827421130209954998234265953843191542003179657... and c = 0.020386712793003585674903530668163000681070027... . - Vaclav Kotesovec, Mar 02 2016

A268300 G.f. satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)), where g.f. A(x) = Sum_{n>=0} a(n)*2*(x/4)^n.

Original entry on oeis.org

1, 7, 119, 2118, 42523, 914922, 20745494, 487390092, 11764545555, 289962708802, 7267069560834, 184626340341588, 4744080078088734, 123075608359376932, 3219261610951795084, 84806249132678044440, 2248017950109054256899, 59917503707743905031346, 1604813748929693765997450, 43170742498490205711682564, 1165893490887496323343495146, 31598783791475055433157814444, 859179326846115018832395000820
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2016

Keywords

Comments

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

Examples

			G.f.: A(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 + 7267069560834*2*x^10/4^10 +...
where g.f. A(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x/A(x))*(1-A(x)) * (1-x^2)*(1-x^2/A(x))*(1-x*A(x)) * (1-x^3)*(1-x^3/A(x))*(1-x^2*A(x)) * (1-x^4)*(1-x^4/A(x))*(1-x^3*A(x)) * (1-x^5)*(1-x^5/A(x))*(1-x^4*A(x)) * (1-x^6)*(1-x^6/A(x))*(1-x^5*A(x)) *...
also
A(x) = 1/((1-x)*(1-x*A(x))*(1-1/A(x)) * (1-x^2)*(1-x^2*A(x))*(1-x/A(x)) * (1-x^3)*(1-x^3*A(x))*(1-x^2/A(x)) * (1-x^4)*(1-x^4*A(x))*(1-x^3/A(x)) * (1-x^5)*(1-x^5*A(x))*(1-x^4/A(x)) * (1-x^6)*(1-x^6*A(x))*(1-x^5/A(x)) *...).
RELATED SERIES.
1/A(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 +...+ A268301(n)/2*x^n/4^n +...
Series_Reversion( x/A(x) ) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...+ A268299(n)*x^n +..., an integer series.
Let J(x) = Sum_{n>=1} x^(n*(n-1)/2) * (A(x)^n + 1/A(x)^(n-1)),
then J(x) is an integer series:
J(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +..+ A268302(n)*x^n +...
and J(x) = Product_{n>=1} (1-x^n) * (1 + x^n/A(x)) * (1 + x^(n-1)*A(x)).
Conjecture: Product_{n>=1} (1-x^n) * (1 + k*x^n/A(x)) * (1 + k*x^(n-1)*A(x)) yields an integer series for all integer k.

Links

Crossrefs

Cf. A268301, A268302, A268299, A190791.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {4/r, 1/(2*Sqrt[2*Pi]) * Sqrt[(r*s^4* Log[r]*(((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r])/(r*s^2) - ((-1 + s)*(-1 + r*s)*Log[r] * Derivative[0, 1][QPochhammer][r, r])/(s^2 * QPochhammer[r, r]) + r*Log[r]*QPochhammer[r, r]*QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + ((-1 + s)*(QPochhammer[s, r]*(Log[r] + (1 - r*s)* QPolyGamma[0, -Log[r*s]/Log[r], r]) + r*(1 - r*s)*Log[r]* Derivative[0, 1][QPochhammer][s, r]))/(r*s^2 * QPochhammer[s, r]))) / (2*Log[r]^2 + (-3 + s*(1 + r + r*s)) * Log[r] * QPolyGamma[0, Log[s]/Log[r], r] + (-1 + s)*(-1 + r*s) * QPolyGamma[0, Log[s]/Log[r], r]^2 + ((3 - s*(1 + r + r*s))*Log[r] - 2*(-1 + s)*(-1 + r*s) * QPolyGamma[0, Log[s]/Log[r], r]) * QPolyGamma[0, -Log[r*s]/Log[r], r] + (-1 + s)*(-1 + r*s) * QPolyGamma[0, -Log[r*s]/Log[r], r]^2 + (-1 + s)*(-1 + r*s)* QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)*(-1 + r*s)* QPolyGamma[1, -Log[r*s]/Log[r], r])]} /. FindRoot[{(1 - 1/(r*s))*(1 - s)/(QPochhammer[r] * QPochhammer[1/(r*s), r] * QPochhammer[s, r]) == s, (-2 + s + r*s)*Log[r] + (-1 + s)*(-1 + r*s)*(QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s]/Log[r], r]) == 0}, {r, 1/7}, {s, 4}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
  • PARI
    {a(n) = my(A=2+x,t=floor(sqrt(2*n+1)+1/2)); for(i=0,n, A = (A + 1/sum(m=-t,t, x^(m*(m+1)/2) * (-A)^m +x*O(x^n)) )/2 ); 4^n/2 * polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

Given g.f. A(x) = Sum_{n>=0} a(n) * 2*(x/4)^n, then g.f. also satisfies:
(1) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n,
(2) A(x) = 1 / Product_{n>=1} (1-x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)),
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(4) x = Sum_{n>=1} A268299(n) * x^n / A(x)^n.
a(n) is odd iff n = 2^k for k>=0 or n=0 (conjecture).
a(n) ~ c * d^n / n^(3/2), where d = 29.10159109069361717048796233905065832... and c = 0.57417747020768285925989822148605305... . - Vaclav Kotesovec, Mar 02 2016
Formula (2) can be rewritten as the functional equation y = 1 / (QPochhammer(x) * QPochhammer(y,x) / (1-y) * QPochhammer(1/(x*y),x) / (1 - 1/(x*y))). - Vaclav Kotesovec, Jan 19 2024

A268302 G.f.: Sum_{n>=1} x^(n*(n-1)/2) * (G(x)^n + 1/G(x)^(n-1)), where G(x) is the g.f. of A268300.

Original entry on oeis.org

3, 8, 28, 144, 736, 4024, 22912, 134784, 813476, 5010904, 31379808, 199196320, 1278911808, 8290414024, 54186864896, 356711621984, 2362968349568, 15739688709864, 105357470567228, 708338644347808, 4781146692837856, 32387329985982176, 220104493513881920, 1500273861724289984, 10253983269166864256, 70258772726034956688, 482514972838806347776, 3320848006096569464080, 22900703924095461843008, 158216154716853989543080
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2016

Keywords

Examples

			G.f.: A(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +...
such that
A(x) = Sum_{n>=1} x^(n*(n-1)/2) * (G(x)^n + 1/G(x)^(n-1)),
that is,
A(x) = (G(x) + 1) + x*(G(x)^2 + 1/G(x)) + x^3*(G(x)^3 + 1/G(x)^2) + x^6*(G(x)^4 + 1/G(x)^3) + x^10*(G(x)^5 + 1/G(x)^4) + x^15*(G(x)^6 + 1/G(x)^5) +...,
where
G(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 +...+ A268300(n)*2*x^n/4^n +...
satisfies:
-1 = Product_{n>=1} (1-x^n) * (1 - x^n/G(x)) * (1 - x^(n-1)*G(x)).

Crossrefs

Cf. A268300, A268301.

Formula

G.f.: Product_{n>=1} (1-x^n) * (1 + x^n/G(x)) * (1 + x^(n-1)*G(x)), where G(x) is the g.f. of A268300.
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