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.

A370145 Expansion of ( (1 + x)*(1 - 2*x)*(1 + 4*x) )^(1/3).

Original entry on oeis.org

1, 1, -3, 3, -12, 30, -102, 318, -1083, 3657, -12747, 44715, -159222, 571332, -2068608, 7538664, -27646374, 101915850, -377496030, 1404077790, -5242135728, 19637862132, -73793090676, 278068062756, -1050503580534, 3977985415746, -15096209345958, 57403753019238, -218683959367908
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Comments

The cube root of F(x) = (1 + x)*(1 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 - 12747*x^10 + 44715*x^11 - 159222*x^12 + ...
where A(x)^3 = (1 + 3*x - 6*x^2 - 8*x^3).
RELATED SERIES.
The following infinite product equals the g.f. of A370015:
A(x)^3 * A(2*x^2)^3 * A(4*x^3)^3 * A(8*x^4)^3 * ... * A(2^(n-1)*x^n)^3 * ... = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ... by the Jacobi triple product identity.
If A(x) = 1/B(x/A(x)) then B(x) = 1/A(x/B(x)) begins
B(x) = 1 - x + 3*x^2 - 9*x^6 + 27*x^8 - 324*x^12 + 1215*x^14 - 18711*x^18 + 75816*x^20 - 1301265*x^24 + 5484996*x^26 - 100048689*x^30 + 431943435*x^32 - 8192222064*x^36 + ... + A370146(n)*x^n + ...

Links

Crossrefs

Cf. A370015, A370016, A370146, A370147, A370149.

Programs

  • Mathematica
    CoefficientList[Series[Surd[(1+x)(1-2x)(1+4x),3],{x,0,30}],x] (* Harvey P. Dale, Oct 04 2024 *)
  • PARI
    {a(n) = polcoeff( (1 + 3*x - 6*x^2 - 8*x^3 +x*O(x^n))^(1/3), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = (1 + x)*(1 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3).
(2) Product_{n>=1} A( 2^(n-1)*x^n )^3 = Sum_{n>=0} 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2), which is the g.f. of A370015.
(3) A(x) = 1/B(x/A(x)) where B(x) = 1/A(x/B(x)) = x/Series_Reversion(x/A(x)) equals the g.f. of A370146.
a(n) ~ (-1)^(n+1) * 2^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 23 2024

A370149 Expansion of ( (1 + x)*(1 - 11*x)*(1 + 121*x) )^(1/3).

Original entry on oeis.org

1, 37, -1776, 114096, -9165936, 810646320, -76152738288, 7450371782832, -750608233752432, 77319392827405872, -8104270335592602864, 861419406835986019248, -92621128795282877608560, 10055062260891607562940720, -1100545944769838408566122480, 121306087657061323164937678512
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2024

Keywords

Comments

The cube root of F(x) = (1 + x)*(1 - 11*x)*(1 + 121*x) = (1 + 111*x - 1221*x^2 - 1331*x^3) has integer coefficients because F(x) == (1+x)^3 (mod 9).

Examples

			G.f.: A(x) = 1 + 37*x - 1776*x^2 + 114096*x^3 - 9165936*x^4 + 810646320*x^5 - 76152738288*x^6 + 7450371782832*x^7 - 750608233752432*x^8 + ...
where A(x)^3 = (1 + 111*x - 1221*x^2 - 1331*x^3).
RELATED SERIES.
We have the following infinite product
A(x)^3 * A(11*x^2)^3 * A(11^2*x^3)^3 * A(11^3*x^4)^3 * ... = 1 + 111*x + 147631*x^3 + 2161452161*x^6 + 348104014265601*x^10 + 616687495357008127151*x^15 + ... + 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370145, A370147, A370334.

Programs

  • PARI
    {a(n) = polcoeff( ( (1 + x)*(1 - 11*x)*(1 + 121*x) +x*O(x^n))^(1/3), n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = (1 + x)*(1 - 11*x)*(1 + 121*x) = (1 + 111*x - 1221*x^2 - 1331*x^3).
(2) Product_{n>=1} A( 11^(n-1)*x^n )^3 = Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2).
a(n) ~ (-1)^(n+1) * 2^(5/3) * 5^(1/3) * 11^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 25 2024

A370148 Expansion of A(x) = [ Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, 19, -361, 4896, -186048, 6361181, -265706784, 10569322565, -439680983904, 18480280546656, -790074277452000, 34174424338394976, -1494143747622128305, 65898152303725266336, -2928713377590693411552, 131019840536990930329051, -5895300394280706457304448, 266614701826937350737301056
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Examples

			G.f.: A(x) = 1 + 19*x - 361*x^2 + 4896*x^3 - 186048*x^4 + 6361181*x^5 - 265706784*x^6 + 10569322565*x^7 - 439680983904*x^8 + 18480280546656*x^9 + ...
The cube of g.f. A(x) equals the infinite product
A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 7^2*x) * (1 - 7*x^2)*(1 - 7^2*x^2)*(1 - 7^3*x^2) * (1 + 7^2*x^3)*(1 + 7^3*x^3)*(1 + 7^4*x^3) * (1 - 7^3*x^4)*(1 - 7^4*x^4)*(1 - 7^5*x^4) * ...
Notice that the cube of A(x) yields the series
A(x)^3 = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 - 76664422756665399911143*x^21 + ... + (-7)^(n*(n-1)/2)*(1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370147, A370016, A370334, A370019, A370336.

Programs

  • PARI
    {a(n) = my(A);
A = prod(m=1, n+1, (1 + (-7)^(m-1)*x^m) * (1 - (-7)^m*x^m) * (1 + (-7)^(m+1)*x^m) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + (-7)^(n-1)*x^n) * (1 - (-7)^n*x^n) * (1 + (-7)^(n+1)*x^n), by the Jacobi triple product identity.
(3) A(x) = Product_{n>=1} F( 2^(n-1)*x^n ), where F(x) = (1 + 57*x + 399*x^2 + 343*x^3)^(1/3) which is the g.f. of A370147.
a(n) ~ (-1)^(n+1) * c * 7^(2*n) / n^(4/3), where c = 0.2168488573077459727164856825904737112... - Vaclav Kotesovec, Feb 24 2024
Showing 1-3 of 3 results.

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