A104996 -id:A104996 - cp's OEIS frontend

A104997 Denominators of coefficients in a series solution to a certain differential equation.

1, 8, 128, 15360, 3440640, 247726080, 653996851200, 476109707673600, 457065319366656000, 43034457761906688000, 850360885375276154880000, 1571466916173510334218240000, 693959790182222163590774784000, 9021477272368888126680072192000000, 27280947271643517695080538308608000000

Offset: 1

Zak Seidov, Mar 31 2005

Series solution of o.d.e. (A. Gruzinov, 2005): cos(t)*f'(t) + sin(t)*f''(t) + (3/4)*sin(t)*f(t) = 0, f(-Pi/2) = 1, f'(-Pi/2) = 0, f(t) = 1 - (3/8)*(t + Pi/2)^2 - (5/128)*(t + Pi/2)^4 - (193/15360)*(t + Pi/2)^4 - ... All coefficients (except 1) are negative, there is no simple recursion or other formula for the series coefficients.

Robert Israel, Table of n, a(n) for n = 1..203
Andrei Gruzinov, Power of an axisymmetric pulsar, Physical Review Letters, Vol. 94, No. 2 (2005), 021101; arXiv preprint, arXiv:astro-ph/0407279, 2004.

Cf. A104996 (numerators).

de:= sin(s)*D(g)(s)-cos(s)*(D@@2)(g)(s)-3/4*cos(s)*g(s)=0:
S:= dsolve({de, g(0)=1, D(g)(0)=0},g(s), series, order=51):
seq(denom(coeff(rhs(S),s,2*j)),j=0..25); # Robert Israel, Jun 05 2019

CoefficientList[Series[Hypergeometric2F1[-1/4, 3/4, 1/2, Sin[x]^2], {x, 0, 30}], x][[1 ;; -1 ;; 2]] // Denominator (* Amiram Eldar, Apr 29 2023 *)

The solution to the o.d.e. is hypergeom([-1/4,3/4],[1/2],sin(t+Pi/2)). - Robert Israel, Jun 05 2019

More terms from Robert Israel, Jun 05 2019

A105372 Decimal expansion of Hypergeometric2F1[ -(1/4),3/4,1,1] = sqrt(Pi)/(Gamma[1/4]*Gamma[5/4]).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Apr 02 2005

This constant appears in solution to an ODE considered in A104996, A104997.

			0.53935260118837935666793572235555273276586896544304013033994...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Andrei Gruzinov, Power of an axisymmetric pulsar, Physical Review Letters, Vol. 94, No. 2 (2005), 021101, preprint, arXiv:astro-ph/0407279, 2004.

Cf. A093341, A104996, A104997.

evalf(1/EllipticK(1/sqrt(2)),120); # Vaclav Kotesovec, Jun 15 2015

RealDigits[1/EllipticK[1/2],10,120][[1]] (* Vaclav Kotesovec, Jun 15 2015 *)

sqrt(Pi)/(gamma(1/4)*gamma(5/4)) \\ G. C. Greubel, Jan 09 2017

Hypergeometric2F1[ -(1/4), 3/4, 1, 1] = Sqrt[Pi]/(Gamma[1/4]*Gamma[5/4]).
From Vaclav Kotesovec, Jun 15 2015: (Start)
4*sqrt(Pi)/Gamma(1/4)^2.
1 / EllipticK(1/sqrt(2)) (Maple notation).
1 / EllipticK[1/2] (Mathematica notation).
(End)
Equals Product_{k>=1} (1 + (-1)^k/(2*k)). - Amiram Eldar, Aug 26 2020

Last digit corrected by Vaclav Kotesovec, Jun 15 2015

A104241 Decimal expansion of the first zero of the solution to the differential equation cos(t)f'(t) + sin(t)f''(t) + (3/4)sin(t)f(t) = 0, f(-Pi/2) = 1, f'(-Pi/2) = 0 (negated).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Apr 02 2005

Andrei Gruzinov, Power of an axisymmetric pulsar, Physical Review Letters, Vol. 94, No. 2 (2005), 021101; arXiv preprint, arXiv:astro-ph/0407279, 2004.

Cf. A104996, A104997, A105372.

RealDigits[x /. FindRoot[Hypergeometric2F1[-1/4, 3/4, 1/2, Sin[x + Pi/2]^2], {x, -0.2}, WorkingPrecision -> 120], 10, 105][[1]] (* Amiram Eldar, Apr 29 2023 *)

-0.22205656674758060918619921293376334697941...

Offset corrected by Michel Marcus, Nov 12 2019

a(53) corrected and more terms added by Amiram Eldar, Apr 29 2023

cp's OEIS Frontend

A104997 Denominators of coefficients in a series solution to a certain differential equation.

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A105372 Decimal expansion of Hypergeometric2F1[ -(1/4),3/4,1,1] = sqrt(Pi)/(Gamma[1/4]*Gamma[5/4]).

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A104241 Decimal expansion of the first zero of the solution to the differential equation cos(t)f'(t) + sin(t)f''(t) + (3/4)sin(t)f(t) = 0, f(-Pi/2) = 1, f'(-Pi/2) = 0 (negated).

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cp's OEIS Frontend

A104997 Denominators of coefficients in a series solution to a certain differential equation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A105372 Decimal expansion of Hypergeometric2F1[ -(1/4),3/4,1,1] = sqrt(Pi)/(Gamma[1/4]*Gamma[5/4]).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A104241 Decimal expansion of the first zero of the solution to the differential equation cos(t)*f'(t) + sin(t)*f''(t) + (3/4)*sin(t)*f(t) = 0, f(-Pi/2) = 1, f'(-Pi/2) = 0 (negated).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A104241 Decimal expansion of the first zero of the solution to the differential equation cos(t)f'(t) + sin(t)f''(t) + (3/4)sin(t)f(t) = 0, f(-Pi/2) = 1, f'(-Pi/2) = 0 (negated).