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.

A122854 Expansion of phi(q)^2*psi(q)^4 in powers of q where phi(),psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, 26, 48, 73, 120, 170, 208, 290, 360, 384, 528, 651, 656, 842, 960, 960, 1248, 1370, 1360, 1682, 1848, 1898, 2208, 2353, 2320, 2810, 3120, 2880, 3480, 3722, 3504, 4420, 4488, 4224, 5040, 5330, 5208, 5760, 6240, 5905, 6888, 7540, 6736, 7922, 8160, 7680
Offset: 0

Views

Author

Michael Somos, Sep 14 2006

Keywords

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

Crossrefs

A050458(2n+1) = A050470(2n+1) = a(n).
Cf. A152584.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(1/2)]^4 * EllipticTheta[3, 0, q]^2)/16, {q, 0, n}]; Table[A122854[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)
  • PARI
    {a(n)= local(A, p, e, f); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, f=(-1)^(p\2); (p^(2*e+2)-f^(e+1))/(p^2-f)))))}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^18/(eta(x+A)^8*eta(x^4+A)^4), n))}

Formula

Expansion of q^(-1/2)eta(q^2)^18/(eta(q)^8*eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [ 8, -10, 8, -6, ...].
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4), b(p^e) = ((p^2)^(e+1)-(-1)^(e+1))/(p^2+1) if p == 3 (mod 4).
G.f.: Sum_{k>0 odd} k^2*x^k/(1+x^(2k)) = Product_{k>0} (1-x^(2k))^6*(1+x^k)^8/(1+x^(2k))^4.
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^3/24 = 1.291928... (A152584). - Amiram Eldar, Dec 29 2023

A194655 Decimal expansion of Pi*(Pi^2*zeta(3) + 6*zeta(5))/8.

Original entry on oeis.org

7, 1, 0, 2, 1, 1, 7, 0, 7, 9, 0, 0, 0, 1, 6, 8, 6, 1, 5, 8, 9, 7, 3, 0, 6, 0, 0, 0, 4, 1, 7, 9, 8, 3, 2, 8, 7, 1, 5, 9, 8, 6, 7, 3, 6, 9, 3, 4, 6, 8, 1, 7, 5, 9, 1, 2, 8, 2, 1, 7, 6, 5, 8, 7, 4, 8, 3, 1, 0, 2, 8, 8, 8, 4, 5, 9, 0, 2, 2, 5, 0, 0, 4, 2, 8, 7, 4, 5, 8, 3, 2, 6, 8, 9, 2, 7, 0, 4, 8, 3, 7, 3, 0, 5, 6
Offset: 1

Views

Author

Seiichi Kirikami, Aug 31 2011

Keywords

Comments

The absolute value of Integral_{x=0..Pi/2} x^2*(log(2*cos(x)))^3 dx.
The absolute value of d^3/db^3(d^2/da^2(Integral_{x=0..Pi/2} cos(ax)*(2*cos(x))^b dx))).
The absolute value of m=2 and n=3 of (Pi/2)*(d^n/db^n(d^m/da^m(gamma(b+1)/gamma((b+a)/2+1)/gamma((b-a)/2+1)))). [Seiichi Kirikami and Peter J. C. Moses]

Examples

			Equals 7.1021170790001686158...

References

  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.631.9

Crossrefs

Cf. A152584, A193712, A193713, A194655.

Programs

  • Mathematica
    RealDigits[ N[Pi (Pi^2*Zeta(3)+6*Zeta(5))/8, 150]][[1]]

Formula

Equals A000796*(A002388*A002117 + 6*A013663)/8.

A193712 Decimal expansion of Pi*zeta(3)/4.

Original entry on oeis.org

9, 4, 4, 0, 9, 3, 2, 8, 4, 0, 4, 0, 7, 6, 9, 7, 3, 1, 8, 0, 0, 8, 6, 8, 9, 4, 8, 3, 1, 3, 1, 3, 5, 7, 0, 5, 3, 7, 5, 3, 0, 7, 5, 9, 3, 1, 9, 9, 1, 6, 3, 3, 2, 4, 3, 9, 5, 7, 3, 8, 3, 1, 0, 7, 2, 1, 1, 3, 8, 6, 6, 3, 7, 5, 6, 6, 2, 5, 0, 8, 2, 9, 4, 6, 4, 1, 9, 6, 0, 5, 6, 6, 6, 4, 8, 9, 6, 7, 6, 6, 3, 6, 4, 7, 5
Offset: 0

Views

Author

Seiichi Kirikami, Aug 31 2011

Keywords

Comments

The absolute value of Integral_{x=0..Pi/2} x^2*log(2*cos(x)) dx.
The absolute value of (d/db(d^2/da^2(Integral_{x=0..Pi/2} cos(ax)*(2*cos(x))^b dx))).
The absolute value of (Pi/2)*(d/db(d^2/da^2(gamma(b+1)/gamma((b+a)/2+1)/gamma((b-a)/2+1))) at a=0 and b=0. - Seiichi Kirikami and Peter J. C. Moses

Examples

			0.94409328404076973180...

References

  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.631.9

Links

Crossrefs

Cf. A000796, A002117, A152584, A193713, A194655.

Programs

  • Mathematica
    RealDigits[ N[Pi Zeta[3]/4, 150]][[1]]

Formula

Equals A000796*A002117/4.
Equals 2 * Integral_{x=0..1} arcsin(x)^2*arccos(x)/x dx (Kobayashi, 2021). - Amiram Eldar, Jun 23 2023

A193713 Decimal expansion of 11*Pi^5/1440.

Original entry on oeis.org

2, 3, 3, 7, 6, 5, 0, 3, 6, 9, 8, 8, 7, 5, 6, 6, 6, 5, 6, 8, 6, 8, 1, 6, 2, 7, 8, 5, 0, 5, 4, 0, 2, 1, 9, 9, 3, 9, 4, 6, 7, 4, 1, 5, 0, 8, 9, 6, 4, 4, 6, 1, 7, 3, 3, 3, 9, 4, 7, 3, 3, 9, 4, 4, 8, 2, 5, 4, 0, 6, 1, 8, 9, 9, 0, 9, 5, 5, 1, 5, 7, 5, 9, 3, 3, 0, 6, 8, 4, 0, 6, 3, 9, 4, 8, 3, 0, 7, 6, 9, 4, 0, 5, 8, 4
Offset: 1

Views

Author

Seiichi Kirikami, Aug 31 2011

Keywords

Comments

The value of Integral_{x=0..Pi/2} x^2*(log(2*cos(x)))^2 dx.
The absolute value of (d^2/db^2(d^2/da^2(Integral_{x=0..Pi/2} cos(a*x)*(2*cos(x))^b dx.
The value of Pi/2*(d^2/db^2(d^2/da^2(gamma(b+1)/gamma((b+a)/2+1)/gamma((b-a)/2+1)))) at a=0 and b=0. [Seiichi Kirikami and Peter J. C. Moses]

Examples

			2.3376503698875666568...

References

  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.631.9

Links

Crossrefs

Cf. A152584, A193712, A194655.

Programs

  • Mathematica
    RealDigits[ N[11 Pi^5/1440, 150]][[1]]

Formula

Equals 11*A092731/1440.
Showing 1-4 of 4 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.