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 1-4 of 4 results.

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, 1, -1, 9, -18, 44, -54, 350, -1359, 3789, -9585, 42489, -163900, 543474, -1933092, 7499404, -27668718, 100329714, -371138346, 1394575578, -5236658316, 19587163968, -73536845444, 278088068628, -1052804678958, 3985553554074, -15132118280498, 57617112474306, -219680808219216
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2024

Keywords

Comments

Equals the self-convolution cube root of A370015.

Examples

			G.f.: A(x) = 1 + x - x^2 + 9*x^3 - 18*x^4 + 44*x^5 - 54*x^6 + 350*x^7 - 1359*x^8 + 3789*x^9 - 9585*x^10 + 42489*x^11 - 163900*x^12 + 543474*x^13 - 1933092*x^14 + 7499404*x^15 + ...
where the cube of g.f. A(x) yields the series
A(x)^3 = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + 11728213508096*x^36 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ...
The cube of g.f. A(x) also equals the infinite product
A(x)^3 = (1 + x)*(1 - 2*x)*(1 + 2^2*x) * (1 + 2*x^2)*(1 - 2^2*x^2)*(1 + 2^3*x^2) * (1 + 2^2*x^3)*(1 - 2^3*x^3)*(1 + 2^4*x^3) * (1 + 2^3*x^4)*(1 - 2^4*x^4)*(1 + 2^5*x^4) * ...
Equivalently,
A(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) * (1 + 6*x^2 - 24*x^4 - 64*x^6)^(1/3) * (1 + 12*x^3 - 96*x^6 - 512*x^9)^(1/3) * (1 + 24*x^4 - 384*x^8 - 4096*x^12)^(1/3) * (1 + 48*x^5 - 1536*x^10 - 32768*x^15)^(1/3) * ...
where
(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) = 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 + ... + A370145(n)*x^n + ...

Links

Crossrefs

Cf. A370015, A370145, A370148, A370334, A370019, A370336.
Cf. A304961, A370716, A370765.

Programs

  • Mathematica
    Round[CoefficientList[Series[QPochhammer[-2, 2*x]^(1/3) * QPochhammer[-1/2, 2*x]^(1/3) * QPochhammer[2*x]^(1/3)*2^(1/3)/3^(2/3), {x, 0, 30}], x]] (* Vaclav Kotesovec, Feb 23 2024 *)
  • PARI
    {a(n) = my(A);
A = prod(m=1, n+1, (1 + 2^(m-1)*x^m) * (1 - 2^m*x^m) * (1 + 2^(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} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + 2^(n-1)*x^n) * (1 - 2^n*x^n) * (1 + 2^(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 + 3*x - 6*x^2 - 8*x^3)^(1/3).
a(n) ~ c * (-1)^(n+1) * 4^n / n^(4/3), where c = 0.260357663494676514371316... - Vaclav Kotesovec, Feb 23 2024
Radius of convergence r = 1/4 (from Vaclav Kotesovec's formula) and A(r) = ( Sum_{n>=0} (1/2^n + 2*2^n)/3 * 1/2^(n*(n+1)/2) )^(1/3) = ( Product_{n>=1} (1 + 1/2^n)*(1 - 1/4^(n+1)) )^(1/3) = 1.298389210904220681888354941631161162163... - Paul D. Hanna, Mar 07 2024

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

A370018 Expansion of Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

Original entry on oeis.org

1, -12, 0, 176, 0, 0, -2752, 0, 0, 0, 43776, 0, 0, 0, 0, -699392, 0, 0, 0, 0, 0, 11186176, 0, 0, 0, 0, 0, 0, -178962432, 0, 0, 0, 0, 0, 0, 0, 2863333376, 0, 0, 0, 0, 0, 0, 0, 0, -45813071872, 0, 0, 0, 0, 0, 0, 0, 0, 0, 733008101376, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11728125427712
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Comments

Equals the self-convolution cube of A370019.

Examples

			G.f.: A(x) = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 - 178962432*x^28 + 2863333376*x^36 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
RELATED SERIES.
The cube root of g.f. A(x) is an integer series starting as
A(x)^(1/3) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 + ... + A370019(n)*x^n + ...
Also,
A(x)^(1/6) = 1 - 2*x - 10*x^2 - 44*x^3 - 330*x^4 - 2508*x^5 - 21476*x^6 - 185720*x^7 - 1658778*x^8 - 15042060*x^9 - 138464620*x^10 + ...
The expansion of 1/A(x) begins
1/A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + ... + A370045(n)*x^n + ...
Further,
1/A(x)^(1/3) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + ... + A370044(n)*x^n + ...

Crossrefs

Cf. A370019, A370045, A370044, A370015, A370335.

Programs

  • PARI
    {a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n));
polcoeff(H=A, n)}
for(n=0, 66, print1(a(n), ", "))

A370335 Expansion of Sum_{n>=0} 5^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

Original entry on oeis.org

1, 15, 0, 275, 0, 0, 5375, 0, 0, 0, 106875, 0, 0, 0, 0, 2134375, 0, 0, 0, 0, 0, 42671875, 0, 0, 0, 0, 0, 0, 853359375, 0, 0, 0, 0, 0, 0, 0, 17066796875, 0, 0, 0, 0, 0, 0, 0, 0, 341333984375, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6826669921875, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 136533349609375
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2024

Keywords

Comments

Equals the self-convolution cube of A370336.

Examples

			G.f.: A(x) = 1 + 15*x + 275*x^3 + 5375*x^6 + 106875*x^10 + 2134375*x^15 + 42671875*x^21 + 853359375*x^28 + 17066796875*x^36 + 341333984375*x^45 + ...
RELATED SERIES.
The cube root of the g.f. A(x) is an integer series starting as
A(x)^(1/3) = 1 + 5*x - 25*x^2 + 300*x^3 - 3000*x^4 + 34375*x^5 - 426750*x^6 + 5539375*x^7 - 73968750*x^8 + 1010175000*x^9 + ... + A370336(n)*x^n + ...

Crossrefs

Cf. A370015.

Programs

  • PARI
    {a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), 5^m*(2*4^m + 1)/3 * x^(m*(m+1)/2) +x*O(x^n));
polcoeff(H=A, n)}
for(n=0, 66, print1(a(n), ", "))
Showing 1-4 of 4 results.

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