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.

A359920 a(n) = coefficient of x^n in A(x) such that x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

Original entry on oeis.org

1, 1, 6, 29, 137, 690, 3815, 22579, 138353, 862692, 5451339, 34911444, 226475135, 1485571965, 9833401534, 65578882177, 440170565711, 2971402946711, 20161828468803, 137434420403678, 940701180157773, 6462787501335564, 44550102080595910, 308041365014677804, 2135938633975050831
Offset: 0

Views

Author

Paul D. Hanna, Jan 22 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 29*x^3 + 137*x^4 + 690*x^5 + 3815*x^6 + 22579*x^7 + 138353*x^8 + 862692*x^9 + 5451339*x^10 + 34911444*x^11 + 226475135*x^12 + ...
where A = A(x) satisfies the doubly infinite sum
x = ... + x^12*(1/A^9 - A^8) + x^5*(1/A^6 - A^5) + x*(1/A^3 - A^2) + (1 - 1/A) + x^2*(A^3 - 1/A^4) + x^7*(A^6 - 1/A^7) + x^15*(A^9 - 1/A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...
also, by the Watson quintuple product identity,
x = (1-x)*(1-x*A)*(1-1/A)*(1-x*A^2)*(1-x/A^2) * (1-x^2)*(1-x^2*A)*(1-x/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^2/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^3/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...
SPECIFIC VALUES.
A(x) at x = 100/738 diverges.
A(100/739) = 1.680090298639836342808608867776256534712736768391...
A(1/8) = 1.40048762211279862753069563580599076131617792526323...
A(1/9) = 1.28067125711115350114265686789651886973848631068277...

Links

Crossrefs

Cf. A359915, A359916, A359919, A359921, A359924, A359719.
Cf. A359914.

Programs

  • Mathematica
    (* Calculation of constant d: *) With[{k = 1}, 1/r /. FindRoot[{r^3*s^3 * QPochhammer[r] * QPochhammer[1/(r*s^2), r^2] *  QPochhammer[1/(r*s), r] * QPochhammer[s, r] * QPochhammer[s^2/r, r^2] / ((-1 + s)*(-1 + r*s)*(-r + s^2)*(-1 + r*s^2)) == k*r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + (2*s)/(-r + s^2) - 2/(s - r*s^3) + (-QPolyGamma[0, -Log[r*s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s^2]/Log[r^2], r^2] + QPolyGamma[0, Log[s^2/r]/Log[r^2], r^2]) / (s*Log[r]) == 0}, {r, 1/7}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    /* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(x - sum(m=-#A,#A, (Ser(A)^(3*m) - 1/Ser(A)^(3*m+1)) * x^(m*(3*m+1)/2) ),#A-1) ); A[n+1]}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(x - prod(m=1,#A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ),#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.
(1) x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
(2) x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
(3) x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 7.388458151593... and c = 0.36167254645... - Vaclav Kotesovec, Mar 19 2023
Formula (3) can be rewritten as the functional equation x = QPochhammer(x) * QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)) * QPochhammer(y^2/x, x^2)/(1 - y^2/x) * QPochhammer(1/(x*y^2), x^2)/(1 - 1/(x*y^2)). - Vaclav Kotesovec, Jan 19 2024

A359914 a(n) = coefficient of x^n in A(x) such that 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(3*n-1)/2) * A(x)^(3*n) * (1 + x^n*A(x)).

Original entry on oeis.org

1, 2, 4, 30, 154, 1078, 7046, 50766, 364268, 2713444, 20384884, 155954760, 1204192106, 9400024042, 73945396990, 586088682472, 4673927031694, 37484566094970, 302098932029282, 2445538771089012, 19875632898821430, 162118004651048048, 1326658157736876148
Offset: 0

Views

Author

Paul D. Hanna, Jan 23 2023

Keywords

Comments

Conjecture: a(n)/2 == A000041(n) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n, for n >= 1.
Conjecture: a(n) == A361552(n+1) (mod 4) for n >= 0. - Paul D. Hanna, Mar 19 2023

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 30*x^3 + 154*x^4 + 1078*x^5 + 7046*x^6 + 50766*x^7 + 364268*x^8 + 2713444*x^9 + 20384884*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
2 = ... + x^26/A^12*(1 + 1/x^4*A) - x^15/A^9*(1 + 1/x^3*A) + x^7/A^6*(1 + 1/x^2*A) - x^2/A^3*(1 + 1/x^1*A) + x^0*A^0*(1 + x^0*A) - x^1*A^3*(1 + x^1*A) + x^5*A^6*(1 + x^2*A) - x^12*A^9*(1 + x^3*A) + x^22*A^12*(1 + x^4*A) + ... + (-1)^n*x^(n*(3*n+1)/2)*A^(3*n)*(1 + x^n*A) + ...
also, by the Watson quintuple product identity,
2 = (1-x)*(1+1*A)*(1+x/A)*(1-x*A^2)*(1-x/A^2) * (1-x^2)*(1+x*A)*(1+x^2/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1+x^2*A)*(1+x^3/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1+x^3*A)*(1+x^4/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...
SPECIFIC VALUES.
A(x) at x = 100/876 diverges.
A(100/877) = 1.557056751214068970380867667963285879403994350720494...
A(1/9) = 1.450191456209956107571253359997937360795442585014595870...
A(1/10) = 1.324252801492679846747365280925526932201317768972870665...

Links

Crossrefs

Cf. A359920, A359915, A359916, A359919, A359921, A359924.
Cf. A361552, A040051, A000041.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{r^3 * s^3 * QPochhammer[r] * QPochhammer[1/(r*s^2), r^2] * QPochhammer[-1/s, r] * QPochhammer[-s/r, r] *  QPochhammer[s^2/r, r^2] / ((1 + s)*(r + s)*(-r + s^2)*(-1 + r*s^2)) == -2, (-3*r^2 - 2*r*(1 + r)*s + r^3*s^2 - s^4 + 2*r*(1 + r)*s^5 + 3*r*s^6) * Log[r] + (1 + s)*(r + s)*(r - s^2)*(-1 + r*s^2) * (QPolyGamma[0, Log[-1/s]/Log[r], r] + QPolyGamma[0, -1/2 - Log[s]/Log[r], r^2] - QPolyGamma[0, -1/2 + Log[s]/Log[r], r^2] - QPolyGamma[0, Log[-s/r]/Log[r], r]) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
  • PARI
    /* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(2 - sum(m=-#A, #A, (-1)^m * x^(m*(3*m-1)/2) * Ser(A)^(3*m) * (1 + x^m*Ser(A)) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(2 - prod(m=1, #A, (1 - x^m) * (1 + x^(m-1)*Ser(A)) * (1 + x^m/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(3*n-1)/2) * A(x)^(3*n) * (1 + x^n*A(x)).
(2) 2 = Product_{n>=1} (1 - x^n) * (1 + x^(n-1)*A(x)) * (1 + x^n/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 8.762505108391427770669887... and c = 0.25785454119524349137288... - Vaclav Kotesovec, Jan 24 2023
Formula (2) can be rewritten as the functional equation QPochhammer(x) * QPochhammer(-y/x, x)/(1 + y/x) * QPochhammer(-1/y, x)/(1 + 1/y) * QPochhammer(y^2/x, x^2)/(1 - y^2/x) * QPochhammer(1/(x*y^2), x^2)/(1 - 1/(x*y^2)) = 2. - Vaclav Kotesovec, Jan 19 2024

A359915 a(n) = coefficient of x^n in A(x) such that A(x) = Sum_{n=-oo..+oo} (x*A(x))^(n*(3*n+1)/2) * (1/x^(3*n) - x^(3*n+1)).

Original entry on oeis.org

1, 1, 5, 23, 121, 713, 4487, 29374, 197896, 1363770, 9570226, 68156319, 491347930, 3578755113, 26295477075, 194677798065, 1450833583380, 10875262975274, 81940144475296, 620223662770067, 4714016885082577, 35962615212212852, 275282740190267268, 2113705107245941938
Offset: 0

Views

Author

Paul D. Hanna, Jan 22 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 23*x^3 + 121*x^4 + 713*x^5 + 4487*x^6 + 29374*x^7 + 197896*x^8 + 1363770*x^9 + 9570226*x^10 + ...
where A = A(x) satisfies the doubly infinite series
A(x) = ... + (x*A)^12*(x^9 - 1/x^8) + (x*A)^5*(x^6 - 1/x^5) + (x*A)*(x^3 - 1/x^2) + (1 - x) + (x*A)^2*(1/x^3 - x^4) + (x*A)^7*(1/x^6 - x^7) + (x*A)^15*(1/x^9 - x^10) + ... + (x*A)^(n*(3*n+1)/2) * (1/x^(3*m) - x^(3*m+1)) + ...
also, by the Watson quintuple product identity,
A(x) = -x * (1-x*A)*(1-x^2*A)*(1-1/x)*(1-x^3*A)*(1-1/x*A) * (1-x^2*A^2)*(1-x^3*A^2)*(1-A)*(1-x^5*A^3)*(1-x*A^3) * (1-x^3*A^3)*(1-x^4*A^3)*(1-x*A^2)*(1-x^7*A^5)*(1-x^3*A^5) * (1-x^4*A^4)*(1-x^5*A^4)*(1-x^2*A^3)*(1-x^9*A^7)*(1-x^5*A^7) * ...

Links

Crossrefs

Cf. A359920, A359916, A359919, A359921, A359924.
Cf. A359914.

Programs

  • PARI
    /* Using the doubly infinite series */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A) - sum(m=-#A, #A, (x*Ser(A))^(m*(3*m+1)/2) * (1/x^(3*m) - x^(3*m+1)) ),#A-2) ); A[n+1]}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(Ser(A) + x*prod(m=1,#A, (1 - x^m*Ser(A)^m) * (1 - x^(m+1)*Ser(A)^m) * (1 - x^(m-2)*Ser(A)^(m-1)) * (1 - x^(2*m+1)*Ser(A)^(2*m-1)) * (1 - x^(2*m-3)*Ser(A)^(2*m-1))),#A-2) ); 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.
(1) A(x) = Sum_{n=-oo..+oo} (x*A(x))^(n*(3*n+1)/2) * (1/x^(3*n) - x^(3*n+1)).
(2) A(x) = -x * Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n+1)*A(x)^n) * (1 - x^(n-2)*A(x)^(n-1)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Watson quintuple product identity.

A359916 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).

Original entry on oeis.org

1, 1, 7, 48, 349, 2718, 22403, 192375, 1701544, 15389227, 141643233, 1322344998, 12491424723, 119177917679, 1146750961711, 11115577075944, 108437559699613, 1063849149587086, 10489551647580027, 103891138998923739, 1033113794091793406, 10310925888014393461
Offset: 1

Views

Author

Paul D. Hanna, Jan 22 2023

Keywords

Examples

			G.f.: A(x) = x + x^2 + 7*x^3 + 48*x^4 + 349*x^5 + 2718*x^6 + 22403*x^7 + 192375*x^8 + 1701544*x^9 + 15389227*x^10 + ...
where A = A(x) satisfies the doubly infinite series
A(x) - 1 = ... + x^12*(A^9 - 1/A^8) + x^5*(A^6 - 1/A^5) + x*(A^3 - 1/A^2) + (1 - A) + x^2*(1/A^3 - A^4) + x^7*(1/A^6 - A^7) + x^15*(1/A^9 - A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...
also, by the Watson quintuple product identity,
-1 = (1-x^1)*(1-x^1*A)*(1-x^1/A)*(1-x^1*A^2)*(1-x^1/A^2) * (1-x^2)*(1-x^2*A)*(1-x^2/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^3/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^4/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...

Links

Crossrefs

Cf. A359915, A359920, A359919, A359921, A359924.
Cf. A359914.

Programs

  • PARI
    /* Using the doubly infinite series */
{a(n) = my(A=[0,1,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 - Ser(A) + sum(m=-#A, #A, x^(m*(3*m+1)/2) * (1/Ser(A)^(3*m) - Ser(A)^(3*m+1)) ),#A-3) ); A[n+1]}
for(n=1,30, print1(a(n),", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(A=[0,1,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 + prod(m=1,#A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^m/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2)),#A-3) ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).
(2) -1 = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^n/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.

A359719 a(n) = coefficient of x^n/n! in A(x) = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)).

Original entry on oeis.org

1, -11, 58, -225, 2146, -14821, 85590, -1974433, 9180658, 2927259, -85838114, 63964584095, -520091681238, 16934937109019, -384678052715594, 5238404820228159, -295855770548974622, 4600244140822151099, -186350295911412573810, 4851711966859680480959
Offset: 1

Views

Author

Paul D. Hanna, Jan 22 2023

Keywords

Examples

			E.g.f.: A(x) = x - 11*x^2/2! + 58*x^3/3! - 225*x^4/4! + 2146*x^5/5! - 14821*x^6/6! + 85590*x^7/7! - 1974433*x^8/8! + 9180658*x^9/9! + 2927259*x^10/10! + ...
where A(x) equals the doubly infinite series
A(x) = ... + x^12*(exp(-9*x) - exp(8*x)) + x^5*(exp(-6*x) - exp(5*x)) + x*(exp(-3*x) - exp(2*x)) + (1 - exp(-x)) + x^2*(exp(3*x) - exp(-4*x)) + x^7*(exp(6*x) - exp(-7*x)) + x^15*(exp(9*x) - exp(-10*x)) + ... + x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)) + ...
also, by the Watson quintuple product identity,
A(x) = (1-x)*(1-x*exp(x))*(1-1*exp(-x))*(1-x*exp(2*x))*(1-x*exp(-2*x)) * (1-x^2)*(1-x^2*exp(x))*(1-x*exp(-x))*(1-x^3*exp(2*x))*(1-x^3*exp(-2*x)) * (1-x^3)*(1-x^3*exp(x))*(1-x^2*exp(-x))*(1-x^5*exp(2*x))*(1-x^5*exp(-2*x)) * (1-x^4)*(1-x^4*exp(x))*(1-x^3*exp(-x))*(1-x^7*exp(2*x))*(1-x^7*exp(-2*x)) * ...

Links

Crossrefs

Cf. A359920, A359919.

Programs

  • PARI
    /* Using the doubly infinite series */
{a(n) = my(X=x+x*O(x^n),M=sqrtint(2*n)); n! * polcoeff( sum(m=-M,M, x^(m*(3*m+1)/2) * (exp(3*m*X) - exp(-(3*m+1)*X)) ), n)}
for(n=1,30, print1(a(n),", "))
  • PARI
    /* Using the quintuple product */
{a(n) = my(X=x+x*O(x^n)); n! * polcoeff( prod(m=1,n, (1 - x^m) * (1 - x^m*exp(X)) * (1 - x^(m-1)*exp(-X)) * (1 - x^(2*m-1)*exp(2*X)) * (1 - x^(2*m-1)*exp(-2*X)) ),n)}
for(n=1,30, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined by the following.
(1) A(x) = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)).
(2) A(x) = Product_{n>=1} (1 - x^n) * (1 - x^n*exp(x)) * (1 - x^(n-1)*exp(-x)) * (1 - x^(2*n-1)*exp(2*x)) * (1 - x^(2*n-1)*exp(-2*x)), by the Watson quintuple product identity.
(3) A(x) = 2*exp(-x/2) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * sinh((6*n+1)*x/2).
(4) A(x) = (1 - exp(-x)) * Product_{n>=1} (1 - x^n) * (1 - 2*x^n*cosh(x) + x^(2*n)) * (1 - 2*x^(2*n-1)*cosh(2*x) + x^(4*n-2)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.