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.

A005148 Sequence of coefficients arising in connection with a rapidly converging series for Pi.

Original entry on oeis.org

0, 1, 47, 2488, 138799, 7976456, 467232200, 27736348480, 1662803271215, 100442427373480, 6103747246289272, 372725876150863808, 22852464771010647496, 1405886026610765892544, 86741060172969340021952
Offset: 0

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Author

Simon Plouffe and N. J. A. Sloane

Keywords

Comments

The paper by Newman and Shanks has an appendix by Don Zagier which eventually leads to an efficient recursive algorithm for the series itself, whereas the main paper treats each term in isolation, which is enormously slower. Using Zagier's appendix one may compute 1000 terms in 25 seconds running PARI/GP on a 500MHz Alpha. - David Broadhurst, Jun 17 2002 (see second version of PARI code here)
Conjecture: The following 2 definitions give the same sequence: (1) numbers k such that 8^m is the highest power of 2 dividing a(k), and (2) numbers k such that k has exactly (m+1) 1's in its binary representation. A018900 is the special case m=1. - Benoit Cloitre, Jun 22 2002, edited by Hugo Pfoertner, Aug 21 2021
Conjecture: There are polynomials P_k(x) such that P_k(m) = the constant term of j_m(tau)^k where j_m is modular for the Hecke group G(lambda_m), j_3 is the Klein invariant j with constant term 744, and P_k(x) = a(k+1) times a product of monic polynomials. - Barry Brent, Nov 25 2022

Examples

			G.f. = x + 47*x^2 + 2488*x^3 + 138799*x^4 + 7976456*x^5 + 467232200*x^6 + ...

References

  • F. Beukers, Letter to D. Shanks, Mar 13 1984
  • J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195; see Exercise 6(a).
  • D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 1985, p. 255-7,276
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005149, A076657, A018900.
CF. A060236 (reduced mod 3).

Programs

  • Mathematica
    a[n_] := a[n]=(Binomial[2n, n](16^n-Binomial[2n, n]^2))/24-Sum[Binomial[2n-2i, n-i]^3a[i], {i, 0, n-1}]
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ ComposeSeries[ Series[ ((Pi / (2 EllipticK[m]))^2 / (1 - 2 m) - 1) / 24, {m, 0, n}], InverseSeries[ Series[ (1 - m) m/16, {m, 0, n}]]], {m, 0, n}]]; (* Michael Somos, Jul 06 2014 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ ComposeSeries[ Series[ ((Pi / (2 EllipticK[m]))^2 / (1 + m) - 1) / 24, {m, 0, n}], InverseSeries[ Series[ -(1 - m)^-2 m/16, {m, 0, n}]]], {m, 0, n}]]; (* Michael Somos, Jul 06 2014 *)
  • PARI
    {a(n) = if( n<1, 0, polcoeff( prod( k=1, (n+1)\2, 1 + x^(2*k - 1), 1 + x *O(x^n))^(24*n), n) / 24)};
  • PARI
    {nt=1000; a=[1]; b=[1]; d=1; e=0; g=0; print(1); for(n=2,nt, c=48*(a[n-1]+g)+128*(d-32*e); e=d; d=c; i=(n-1)\2; g=12*if(n%2==0,a[n/2]^2)+24*sum(j=1,i,a[j]*a[n-j]); h=12*if(n%2==0,b[n/2]^2)+24*sum(j=1,i,b[j]*b[n-j]); f=(c+5*h)/n^2-g; a=concat(a,f); b=concat(b,n*f); print(f))} /* Broadhurst 2002 */
  • PARI
    {a(n)=if(n<1,0,va[n])} {b(n)=n*a(n)} {doit(nt)= local(c,d,e,g); va=vector(nt); va[1]=1; d=1; e=0; g=0; for(n=2,nt, c=48*(a(n-1)+g)+128*(d-32*e); e=d; d=c; g=12*if(n%2==0,a(n/2)^2)+24*sum(j=1,(n-1)\2,a(j)*a(n-j)); va[n]=(c+5*(12*if(n%2==0,b(n/2)^2)+24*sum(j=1,(n-1)\2,b(j)*b(n-j))))/n^2-g; )}; /* Michael Somos, Nov 05 2002 */
  • PARI
    {a(n) = local(an, cb); if( n<1, 0, an = cb = vector(n, i, binomial(2*i, i)); an[1]=1; for(j=2, n, an[j] = (cb[j]*16^j - cb[j]^3) / 24 - sum(i=1, j-1, cb[j-i]^3*an[i])); an[n])}; /* Michael Somos, Mar 09 2004 */

Formula

a(n) = (1/24) * coefficient of x^n in Product_{k>=1} (1+x^(2k-1))^(24n).
Asymptotically (D. Zagier): a(n) = C*(64^n)/sqrt(n)*(1 - a/n + b/n^2 + ...) with C = (sqrt(Pi)/12)*Gamma(3/4)^2/Gamma(1/4)^2 = 0.0168732651....; a = 6*Gamma(3/4)^4/Gamma(1/4)^4 = 0.078300067..., b = 60*Gamma(3/4)^8/Gamma(1/4)^8 - 1/128 = 0.002405668.... - Benoit Cloitre, Jun 22 2002; numerical value of constant "a" corrected by Vaclav Kotesovec, Jul 28 2013
Alternative expressions for these constants: C = Pi^(5/2)/(6*Gamma(1/4)^4), a = 24*Pi^4/Gamma(1/4)^8, b = 960*Pi^8/Gamma(1/4)^16 - 1/128. - Vaclav Kotesovec, Jul 28 2013
A076657(n) = Sum_{i=0..n} binomial(2*n-2*i, n-i)^3 a(i) = (1/24)*binomial(2*n, n)*(16^n-binomial(2*n, n)^2) (Shanks and Beukers). - Ralf Stephan, Oct 24 2002
Expansion of ((Pi / (2 K(q)))^2 / (1 - 2*k(q)^2) - 1) / 24 in powers of (k'(q) * k(q) / 4)^2. [Borwein and Borwein, 6(a)(i)] - Michael Somos, Jul 06 2014
Expansion of ((Pi / (2 K(q)))^2 / (1 + k(q)^2) - 1) / 24 in powers of (k'(q)^-2 * k(q) / 4)^2. [Borwein and Borwein, 6(a)(ii)] - Michael Somos, Jul 06 2014

Extensions

More terms from Michael Somos, Nov 24 2001

A014103 Expansion of (eta(q^2) / eta(q))^24 in powers of q.

Original entry on oeis.org

1, 24, 300, 2624, 18126, 105504, 538296, 2471424, 10400997, 40674128, 149343012, 519045888, 1718732998, 5451292992, 16633756008, 49010118656, 139877936370, 387749049720, 1046413709980, 2754808758144, 7087483527072, 17848133716832, 44056043512488, 106727749011456
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(q), Greenhill (1895) denotes -64 * A(q^2) by tau_0 on page 409 equation (43). - Michael Somos, Jul 17 2013

Examples

			G.f. = q + 24*q^2 + 300*q^3 + 2624*q^4 + 18126*q^5 + 105504*q^6 + 538296*q^7 + ...

References

  • John H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
  • Albert Eagle, Elliptic functions as they should be, Galloway and Porter Ltd., Cambridge, pp. 72-73.

Links

Crossrefs

Cf. A000594, A005149, A007191, A115977.

Programs

  • Maple
    q*mul((1+q^m)^24,m=1..30); seq(coeff(series(%,q,n+1),q,n), n=1..25);
  • Mathematica
    a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^-24, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ q / Product[ 1 - q^k, {k, 1, n + 1, 2}]^24, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ (m/16)^2 / (1 - m), {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m/16)^2 /(1 - m), {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
eta[q_]:=q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^2] / eta[q])^24, {q, 0, n}]; Table[a[n], {n, 4, 25}] (* Vincenzo Librandi, Oct 18 2018 *)
  • PARI
    {a(n) = polcoeff( x * prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^24, n)};
  • PARI
    {a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst( A, x, x^2); A2 = A * (1 + 16*A); A = 8 * A2 + (1 + 32*A) * sqrt(A2)); polcoeff( A + 16 * A^2, n))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^24, n))};

Formula

REVERT(A005149).
Euler transform of period 2 sequence [ 24, 0, 24, 0, ... ]. - Michael Somos, Mar 19 2004
Expansion of (lambda(q) / 16)^2 / (1 - lambda(q)) in powers of q = exp(2 Pi i t) where lambda() is the elliptic modular function A115977. - Michael Somos, Nov 19 2005
Expansion of q / chi(-q)^24 in powers of q where chi() is a Ramanujan theta function.
Expansion of (theta_2(q) * theta_3(q) / (2 * theta_4(q)^2))^4 = (theta_2(q^(1/2))^2 / (4*theta_4(q^(1/2)) * theta_3(q^(1/2))))^4 in powers of q.
G.f.: x * Product_{k > 0} (1 + x^k)^24 = x / Product_{k > 0} (1 - x^(2*k - 1))^24.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 48*u*v - 4096*u*v^2. - Michael Somos, Mar 19 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = (1/4096) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A007191. - Michael Somos, Aug 19 2007
j(q) = (f(q) + 16)^3 / f(q), j(q^2) = (f(q) + 256)^3 / f(q)^2 where j(q) is the g.f. for A000521 and f(q) is 4096 times the g.f. for a(n). - Michael Somos, Oct 01 2007
Convolution inverse of A007191. Series reversion of A005149.
Sum_{n>=1} exp(-2*Pi*n)*a(n) = 1/512. - Simon Plouffe, Feb 20 2011 [Proof: Sum_{n>=0} a(n)/exp(2*Pi*n) = exp(-2*Pi) * (phi(exp(-4*Pi)) / phi(exp(-2*Pi)))^24 = exp(-2*Pi) * A292821^24, where phi(q) is the Euler modular function. - Vaclav Kotesovec, May 13 2023]
a(n) ~ exp(2 * Pi * sqrt(2*n)) / (4096 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(1) = 1, a(n) = (24/(n-1))*Sum_{k=1..n-1} A000593(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017
G.f.: x*exp(24*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Expansion of Delta(q^2)/Delta(q) in powers of q where the discriminant Delta(q) is the g.f. of A000594. - Michael Somos, May 27 2022
From Vaclav Kotesovec, May 08 2023, updated May 16 2023: (Start)
Sum_{n>=1} a(n) / exp(Pi*n) = exp(-Pi) * A292820^24 = 1/8.
Sum_{n>=1} a(n) / exp(3*Pi*n) = exp(-3*Pi) * A292887^24 = (2 + sqrt(3) - sqrt(9 + 6*sqrt(3)))^8 / 512.
Sum_{n>=1} a(n) / exp(4*Pi*n) = exp(-4*Pi) * A292822^24 = 99*sqrt(2)/2048 - 35/512.
Sum_{n>=1} a(n) / exp(5*Pi*n) = exp(-5*Pi) * A292904^24 = (2 + sqrt(5) - sqrt((15 + 7*sqrt(5))/2))^12 / 8.
Sum_{n>=1} a(n) / exp(6*Pi*n) = exp(-6*Pi) * (A363020/A363018)^24 = 385/512 + 7*sqrt(3)/16 - sqrt(74664 + 43134*sqrt(3))/256.
Sum_{n>=1} a(n) / exp(7*Pi*n) = exp(-7*Pi) * (A363119/A363117)^24 = (sqrt(7) - 1 - sqrt(22*sqrt(7) - 56))^3 / (2^(15/2) * (2^(1/4)*sqrt(5 + sqrt(7)) + (56 + 23*sqrt(7))^(1/4))^6).
Sum_{n>=1} a(n) / exp(8*Pi*n) = exp(-8*Pi) * (A292864/A259151)^24 = -8963/512 - 99*sqrt(2)/8 + 9*sqrt(126913704 + 89741542*sqrt(2))/4096.
Sum_{n>=1} a(n) / exp(9*Pi*n) = exp(-9*Pi) * (A363120/A363118)^24 = ((6*(3 + sqrt(3)))^(1/3) - 3)^8 / (8*((3*(6 + 7*sqrt(3) + 3*sqrt(14*sqrt(3) - 15)))^(1/3) - 3)^8).
Sum_{n>=1} a(n) / exp(10*Pi*n) = exp(-10*Pi) * (A363021/A363019)^24 = (5^(1/4) - 1)^24 / 2097152.
Sum_{n>=1} a(n) / exp(Pi*n/2) = exp(-Pi/2) * A292819^24 = 35 + 99/2^(3/2).
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(Pi*n) = 1/64.
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(2*Pi*n) = 99/2^(3/2) - 35.
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(3*Pi*n) = 97/64 - 7*sqrt(3)/8.
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(4*Pi*n) = -5018696 - 3548754*sqrt(2) + (9/2)*sqrt(2487635528172 + 1759023951091*sqrt(2)).
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(7*Pi*n) = 13880161/64 + 81972*sqrt(7) - 9*sqrt(74328271227 + 28093445864*sqrt(7))/8. (End)
The g.f. A(q) satisfies -(16)^2 * A(q^2) = (lambda(q) + lambda(-q)) = (lambda(q)*lambda(-q)), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus. - Peter Bala, Sep 26 2023
From Peter Bala, Sep 26 2023: (Start)
A(q^2) = -A(q)*A(-q).
A(q) = lambda(-q)^2/(16*lambda(q)) = lambda(-q)*(lambda(-q) - 1)/16. (End)
G.f. A(x) satisfies 0 = f(A(x), A(-x)) where f(u, v) = u + v + 48*u*v - 4096*u^2*v^2. - Michael Somos, Oct 07 2024

Extensions

More terms from Michael Somos, Nov 24 2001

A195130 Series inversion of A100130.

Original entry on oeis.org

1, 24, 852, 35744, 1645794, 80415216, 4094489992, 214888573248, 11542515402255, 631467591949480, 35063515239394764, 1971043639046131296, 111949770626330347638, 6414671157989386260432, 370360217892318010055832
Offset: 1

Views

Author

Michael Somos, Sep 11 2011

Keywords

Examples

			G.f. = q + 24*q^2 + 852*q^3 + 35744*q^4 + 1645794*q^5 + 80415216*q^6 + ...

Links

Crossrefs

Cf. A005149, A100130.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ q / QPochhammer[ -q, q^2]^24, {q, 0, n}], q], {q, 0, n}]];
  • PARI
    {a(n) = if( n<1, 0, polcoeff( serreverse( x * prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n) )^24), n))};

Formula

A005149(n) = -(-1)^n * a(n).
Showing 1-3 of 3 results.

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