A241420 -id:A241420 - cp's OEIS frontend

A019610 Decimal expansion of Pi*e/2.

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

Offset: 1

N. J. A. Sloane

			4.26986711133678353273177543477328724751744426788255748...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 136.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Z. A. Melzak, Infinite products for πe and π/e, Amer. Math. Monthly 68 (1961) 39-41.

Cf. A019609, A241420, A241421.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)*Exp(1)/2; // G. C. Greubel, Aug 24 2018

Digits:=100: evalf(Pi*exp(1)/2); # Wesley Ivan Hurt, Aug 09 2014

RealDigits[(Pi*E)/2,10,120][[1]] (* Harvey P. Dale, Apr 16 2014 *)

{ default(realprecision, 100); x=(1/2)*Pi*exp(1); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 13 2015

Melzak's formula: lim_{n->infinity} Product_{k=1..2n+1} (1+2/k)^(k*(-1)^(k+1)) = Pi*e/2. - Jean-François Alcover, Apr 25 2014

A241421 Decimal expansion of D(1), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 08 2014

			2.23588559550896986428396479931189064484515912285952474779344797826...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 136.

G. C. Greubel, Table of n, a(n) for n = 1..5000
L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (15).
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 20.
Eric Weisstein's MathWorld, Barnes G-Function
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Cf. A006752, A019610 (D(2)), A074962, A241420 (D(1/2)).

RealDigits[Glaisher^6/(2^(1/6)*Sqrt[Pi]), 10, 104] // First

default(realprecision, 100); A=exp(1/12-zeta'(-1)); A^6/(2^(1/6)* sqrt(Pi)) \\ G. C. Greubel, Aug 24 2018

D(x) = lim_{n->infinity} ( Product_{k=1..2n+1} (1+x/k)^((-1)^(k+1)*k) ).
D(x) = (e^(x/2-1/4)*A^3*G((x+1)/2)^2*Gamma(x/2)^(x-2)*Gamma((x+1)/2)^(1-x)*(Gamma((x+1)/2)/Gamma(x/2))^x)/(2^(1/12)*G(x/2)^2), where A is the Glaisher-Kinkelin constant and G is the Barnes G-function.
D(1) = A^6/(2^(1/6)*sqrt(Pi)).

cp's OEIS Frontend

A019610 Decimal expansion of Pi*e/2.

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