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.

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

A370147 Expansion of ( (1 + x)*(1 + 7*x)*(1 + 49*x) )^(1/3).

Original entry on oeis.org

1, 19, -228, 6492, -216372, 7851180, -300848772, 11974587132, -490113592788, 20492868223308, -871404823013412, 37562003034015900, -1637401559515373172, 72053378865932154348, -3196217668534369463748, 142763786831538212246076, -6415201218873454789867284, 289797678008730755585589900
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Comments

The cube root of F(x) = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).
In general, for k > 1, if g.f. = ((1 + x)*(1 + k*x)*(1 + k^2*x))^(1/3), then a(n) ~ (-1)^(n+1) * (k-1)^(2/3) * (k+1)^(1/3) * k^(2*n-1) / (3*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Feb 24 2024

Examples

			G.f.: A(x) = 1 + 19*x - 228*x^2 + 6492*x^3 - 216372*x^4 + 7851180*x^5 - 300848772*x^6 + 11974587132*x^7 - 490113592788*x^8 + ...
where A(x)^3 = (1 + 57*x + 399*x^2 + 343*x^3).
RELATED SERIES.
We have the following infinite product involving the g.f. A(x)
A(x)^3 * A(-7*x^2)^3 * A(49*x^3)^3 * A(-343*x^4)^3 * A(2401*x^5)^3 * ... = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 + ... + (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370145, A370149, A370148.

Programs

  • PARI
    {a(n) = polcoeff( ( (1 + x)*(1 + 7*x)*(1 + 49*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 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3).
(2) Product_{n>=1} A( (-7)^(n-1)*x^n )^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
a(n) ~ (-1)^(n+1) * 2^(5/3) * 7^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 24 2024

A370334 Expansion of [ Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, 37, -1369, 133632, -9888768, 845367083, -78838949376, 7721334144755, -776624602305024, 79868229118115328, -8362877755373222400, 888226662691859185152, -95442299152209579505105, 10355840499178710443340288, -1132966823558169033184762368, 124832961812953439236127605357
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2024

Keywords

Examples

			G.f.: A(x) = 1 + 37*x - 1369*x^2 + 133632*x^3 - 9888768*x^4 + 845367083*x^5 - 78838949376*x^6 + 7721334144755*x^7 - 776624602305024*x^8 + ...
where the cube of g.f. A(x) yields the series
A(x)^3 = 1 + 111*x + 147631*x^3 + 2161452161*x^6 + 348104014265601*x^10 + 616687495357008127151*x^15 + 12017494675541950940487123311*x^21 + ... + 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) + ...
The cube of g.f. A(x) also equals the infinite product
A(x)^3 = (1 + x)*(1 - 11*x)*(1 + 11^2*x) * (1 + 11*x^2)*(1 - 11^2*x^2)*(1 + 11^3*x^2) * (1 + 11^2*x^3)*(1 - 11^3*x^3)*(1 + 11^4*x^3) * (1 + 11^3*x^4)*(1 - 11^4*x^4)*(1 + 11^5*x^4) * ...

Links

Crossrefs

Cf. A370149, A370016, A370148, A370019, A370336.

Programs

  • PARI
    {a(n) = my(A); A = prod(m=1, n+1, (1 + 11^(m-1)*x^m) * (1 - 11^m*x^m) * (1 + 11^(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} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + 11^(n-1)*x^n) * (1 - 11^n*x^n) * (1 + 11^(n+1)*x^n), by the Jacobi triple product identity.
(3) A(x) = Product_{n>=1} F( 11^(n-1)*x^n ), where F(x) = (1 + 111*x - 1221*x^2 - 1331*x^3)^(1/3).
a(n) ~ (-1)^(n+1) * c * 11^(2*n) / n^(4/3), where c = 0.2588865455859866840901787578907966... - Vaclav Kotesovec, Feb 27 2024

A370779 Expansion of ( (1 + x)*(1 - 5*x)*(1 + 25*x) )^(1/3).

Original entry on oeis.org

1, 7, -84, 1020, -17220, 313068, -6075444, 122709468, -2553130020, 54317619660, -1175968479252, 25819593611196, -573476704909572, 12861006710141100, -290799326551852020, 6621725329384239516, -151707434284857934308, 3494405505576163607436
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2024

Keywords

Comments

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

Crossrefs

Cf. A370145, A370149, A370780.
Cf. A370782.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(((1+x)*(1-5*x)*(1+25*x))^(1/3))

A370780 Expansion of ( (1 + x)*(1 - 17*x)*(1 + 289*x) )^(1/3).

Original entry on oeis.org

1, 91, -9828, 1535868, -294731892, 62322050700, -13990450587012, 3270320252339868, -787131217405990548, 193694053976566000812, -48497295306135216560292, 12313491783703337923916220, -3162498663877264843739477172
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2024

Keywords

Comments

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

Crossrefs

Cf. A370145, A370149, A370779.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(((1+x)*(1-17*x)*(1+289*x))^(1/3))
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.