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.

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

A268301 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, -70, -795, -13802, -277782, -6093708, -139376659, -3297234754, -79988099074, -1979248977748, -49758116194846, -1267321717299236, -32631825106297228, -848030793254951704, -22214311484843607811, -585938143786366837938, -15548874443787002057610, -414829266882771282611204, -11120089118043870668697578, -299364678394845043715844268, -8090271856987498430846360564
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) = 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 -...
where g.f. A(x) satisfies the Jacobi Triple Product:
-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)) *...
also
A(x) = (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)) *...
RELATED SERIES.
1/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 +...+ A268300(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) + 1/A(x)^n),
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. A268300, A268302, A268299, A190791.

Programs

  • PARI
    {a(n) = my(A=1/2+x,t=floor(sqrt(2*n+1)+1/2)); for(i=0,n, A = (A + sum(m=-t,t, x^(m*(m-1)/2) * (-A)^m +x*O(x^n)) )/2 ); 2*4^n * 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) = Product_{n>=1} (1-x^n) * (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)),
(3) A(x) = 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-1 for k>=0 (conjecture).
a(n) ~ -c * d^n / n^(3/2), where d = 29.101591090693617170487962339050658... and c = 0.1385938593465955724446602611055779... . - Vaclav Kotesovec, Mar 02 2016

A196354 E.g.f.: 1 + Sum_{n>=1} 2*cosh(n*x) * x^(n^2).

Original entry on oeis.org

1, 2, 0, 6, 48, 10, 2880, 14, 53760, 725778, 645120, 359251222, 6082560, 42032390426, 49201152, 2648040595230, 41845937602560, 115757203161634, 102437981698129920, 3958896348126758, 51901909523009372160, 113368395423628842, 12788630502806158049280
Offset: 0

Views

Author

Paul D. Hanna, Oct 28 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 0*x^2/2! + 6*x^3/3! + 48*x^4/4! + 10*x^5/5! + 2880*x^6/6! + 14*x^7/7! + 53760*x^8/8! +...
The e.g.f. A(x) may be expressed by the series:
A(x) = 1 + 2*cosh(x)*x + 2*cosh(2*x)*x^4 + 2*cosh(3*x)*x^9 + 2*cosh(4*x)*x^16 + 2*cosh(5*x)*x^25 +...
and by Jacobi's triple product:
A(x) = (1-x^2)*(1+x*exp(x))*(1+x/exp(x)) * (1-x^4)*(1+x^3*exp(x))*(1+x^3/exp(x)) * (1-x^6)*(1+x^5*exp(x))*(1+x^5/exp(x)) * (1-x^8)*(1+x^7*exp(x))*(1+x^7/exp(x)) *...

Links

Crossrefs

Cf. A190791, A197707, A192502.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (Series[ EllipticTheta[ 3, x I/2, q], {q, 0, n}] // Normal // TrigToExp) /. {x -> q}, {q, 0, n}]] (* Michael Somos, Nov 18 2011 *)
  • PARI
    {a(n)=local(A=1+x); A=1+sum(m=1,sqrtint(n+1), 2*cosh(m*x+x*O(x^n))*x^(m^2)); n!*polcoeff(A, n)}
  • PARI
    /* By Jacobi's Triple Product Identity: */
{a(n)=local(A=1+x); A=prod(m=1, n\2+1, (1-x^(2*m))*(1+exp(x+x*O(x^n))*x^(2*m-1))*(1+exp(-x+x*O(x^n))*x^(2*m-1)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f.: Product_{n>=1} (1 - x^(2*n))*(1 + x^(2*n-1)*exp(x))*(1 + x^(2*n-1)/exp(x)), due to the Jacobi triple product identity.
E.g.f.: theta_3( i q/2, q ). - Michael Somos, Oct 29 2011

A197707 G.f.: A(x) = 1 + Sum_{n>=1} x^(n^2) * ((1-x)^n + 1/(1-x)^n).

Original entry on oeis.org

1, 2, 0, 1, 3, 1, 5, 5, 6, 9, 8, 18, 19, 26, 33, 41, 52, 60, 87, 99, 132, 166, 209, 261, 323, 398, 481, 604, 716, 893, 1086, 1331, 1629, 1991, 2428, 2952, 3578, 4314, 5217, 6229, 7508, 8967, 10737, 12838, 15345, 18334, 21894, 26127, 31149, 37093, 44100
Offset: 0

Views

Author

Paul D. Hanna, Oct 17 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + x^3 + 3*x^4 + x^5 + 5*x^6 + 5*x^7 + 6*x^8 +...
where the g.f. A(x) may be expressed as the q-series:
A(x) = 1 + x*((1-x) + 1/(1-x)) + x^4*((1-x)^2 + 1/(1-x)^2) + x^9*((1-x)^3 + 1/(1-x)^3) + x^16*((1-x)^4 + 1/(1-x)^4) +...
and the Jacobi triple product:
A(x) = (1-x^2)*(1+x*(1-x))*(1+x/(1-x)) * (1-x^4)*(1+x^3*(1-x))*(1+x^3/(1-x)) * (1-x^6)*(1+x^5*(1-x))*(1+x^5/(1-x)) *...

Links

Crossrefs

Cf. A190791.

Programs

  • PARI
    {a(n)=local(A); A=1+sum(m=1, sqrtint(n)+1, x^(m^2)*((1-x)^m+1/(1-x+x*O(x^n))^m)); polcoeff(A, n)}
  • PARI
    {a(n)=local(A); A=prod(m=1, n\2+1, (1-x^(2*m))*(1+x^(2*m-1)*(1-x))*(1+x^(2*m-1)/(1-x+x*O(x^n)))); polcoeff(A, n)}

Formula

G.f.: A(x) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^(2*n-1)*(1-x)) * (1 + x^(2*n-1)/(1-x)), due to the Jacobi triple product identity.

A200222 G.f. satisfies: A(x) = 1 + Sum_{n>=1} (x*A(x))^(n^2) * (A(x)^n + 1/A(x)^n).

Original entry on oeis.org

1, 2, 4, 12, 42, 164, 688, 3024, 13680, 63110, 295520, 1401012, 6713280, 32470468, 158343504, 777725264, 3843992546, 19104857608, 95419519076, 478668009828, 2410698765472, 12184259877320, 61782045169312, 314202878599696, 1602270787137472, 8191160756085318, 41971595130249968, 215522156779513584
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 164*x^5 + 688*x^6 +...
The g.f. A = A(x) may be expressed by the series:
A(x) = 1 + x*A*(A + 1/A) + x^4*A^4*(A^2 + 1/A^2) + x^9*A^9*(A^3 + 1/A^3) + x^16*A^16*(A^4 + 1/A^4) + x^25*A^25*(A^5 + 1/A^5) +...
and by the Jacobi triple product:
A(x) = (1+x)*(1+x*A^2)*(1-x^2*A^2) * (1+x^3*A^2)*(1+x^3*A^4)*(1-x^4*A^4) * (1+x^5*A^4)*(1+x^5*A^6)*(1-x^6*A^6) * (1+x^7*A^6)*(1+x^7*A^8)*(1-x^8*A^8) *...

Links

Crossrefs

Cf. A190791, A199576, A196354, A197707.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{s == r^2*s^2 * QPochhammer[-1/r, r^2*s^2] * QPochhammer[-1/(r*s^2), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((1 + r)*(1 + r*s^2)), 1 - r*s^2 - 2*(1 + r*s^2) * QPolyGamma[0, 1, r^2*s^2] / Log[r^2*s^2] + 2*(1 + r*s^2) * QPolyGamma[0, Log[-1/(r*s^2)] / Log[r^2*s^2], r^2*s^2] / Log[r^2*s^2] + 2*r^2*s^2*((1 + r*s^2)*(Derivative[0, 1][QPochhammer][-1/r, r^2*s^2] / QPochhammer[-1/r, r^2*s^2] + Derivative[0, 1][QPochhammer][-1/(r*s^2), r^2*s^2] / QPochhammer[-1/(r*s^2), r^2*s^2]) + r^2*s * QPochhammer[-1/r, r^2*s^2] * QPochhammer[-1/(r*s^2), r^2*s^2] *  Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] / (1 + r)) == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1,n,A=1+sum(m=1, sqrtint(n+1), (A^m +1/(A+x*O(x^n))^m)*(x*A)^(m^2))); polcoeff(A, n)}
  • PARI
    /* By Jacobi's Triple Product Identity: */
{a(n)=local(A=1+x); for(i=1,n,A=prod(m=1, n\2+1, (1+x^(2*m-1)*A^(2*m-2)+x*O(x^n))*(1+x^(2*m-1)*A^(2*m))*(1-x^(2*m)*A^(2*m)) )); polcoeff(A, n)}

Formula

By the Jacobi triple product identity, g.f. A(x) satisfies:
(1) A(x) = Product_{n>=1} (1 + x^(2*n-1)*A(x)^(2*n-2)) * (1 + x^(2*n-1)*A(x)^(2*n)) * (1 - x^(2*n)*A(x)^(2*n)).
Let G(x) be the g.f. of A190791, then A(x) satisfies:
(2) A(x) = (1/x)*Series_Reversion(x/G(x)),
(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
(4) a(n) = [x^n] G(x)^(n+1)/(n+1),
where G(x) = 1 + Sum_{n>=1} x^(n^2) * (G(x)^n + 1/G(x)^n).
a(n) ~ c * d^n / n^(3/2), where d = 5.42800145666083947972618... and c = 0.45497910593346577587... - Vaclav Kotesovec, Sep 04 2017

A216878 G.f. satisfies: A(x) = 1 / Product_{n>=1} (1 + x^n*A(x)) * (1 + x^n/A(x)) * (1-x^n).

Original entry on oeis.org

1, -1, 1, -3, 6, -17, 43, -125, 348, -1029, 3020, -9116, 27567, -84620, 260949, -812053, 2539208, -7989121, 25244540, -80136851, 255325972, -816447638, 2618870068, -8425244209, 27176810469, -87879769383, 284813417885, -925013053556, 3010106492409, -9813119711706
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2012

Keywords

Examples

			G.f.: A(x) = 1 - x + x^2 - 3*x^3 + 6*x^4 - 17*x^5 + 43*x^6 - 125*x^7 +...
such that
1/A(x) = (1+x*A(x))*(1-x/A(x))*(1-x) * (1+x^2*A(x))*(1-x^2/A(x))*(1-x^2) * (1+x^3*A(x))*(1-x^3/A(x))*(1-x^3) * (1+x^4*A(x))*(1-x^4/A(x))*(1-x^4) *...
1/A(x) = (A(x) + x/A(x)) + (A(x)^2 + x^2/A(x)^2)*x + (A(x)^3 + x^3/A(x)^3)*x^3 + (A(x)^4 + x^4/A(x)^4)*x^6 + (A(x)^5 + x^5/A(x)^5)*x^10 +...

Crossrefs

Cf. A190791.

Programs

  • PARI
    {a(n)=local(A=1-x);for(i=1,n,A=1/prod(m=1,n,(1+x^m/A)*(1+x^m*A)*(1-x^m)+x*O(x^n)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1-x);for(i=1,n,A=1/2*(A+1/sum(m=1,sqrtint(8*n+1),(A^m+x^m/A^m)*x^(m*(m-1)/2)+x*O(x^n))));polcoeff(A,n)}
for(n=0,21,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 / Sum_{n>=1} (A(x)^n + x^n/A(x)^n) * x^(n*(n-1)/2) due to the Jacobi triple product identity.
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