The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A161149 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A161149

a(n) = (2*n)!*(2*n+1)!/n! = n!*A000909(n), n=0,1...
(history; published version)

#12 by Charles R Greathouse IV at Thu Sep 08 08:45:45 EDT 2022

PROG

(MAGMAMagma) [Factorial(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 06 2015

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#11 by Jon E. Schoenfield at Tue Jul 07 21:50:17 EDT 2015

STATUS

editing

approved

#10 by Jon E. Schoenfield at Tue Jul 07 21:50:15 EDT 2015

COMMENTS

W(x)=1/2*(-1/2*Pi*x^(1/2)*hypergeom([],[1/2, 3/2],-1/16*x)+Pi^(1/2)-1/16*Pi^(1/2)*x*sum((-1)^(2*j)* (-Psi(3/2+j)+Pi*tan(Pi*j)-Psi(2+j)-Psi(1+j)-4*lnlog(2)+lnlog(x))*2^(-4*j)* (x^j)*sec(Pi*j)*2^(2*j)/(1/2+j)/GAMMA(1+2*j)/GAMMA(2+j), j = 0..infinity))/(x^(1/2)*Pi);

STATUS

approved

editing

#9 by N. J. A. Sloane at Mon Jul 06 23:35:25 EDT 2015

STATUS

proposed

approved

#8 by Vincenzo Librandi at Mon Jul 06 03:11:15 EDT 2015

STATUS

editing

proposed

#7 by Vincenzo Librandi at Mon Jul 06 03:11:03 EDT 2015

DATA

1, 12, 1440, 604800, 609638400, 1207084032000, 4142712397824000, 22619209692119040000, 184572751087691366400000, 2146211949647675208499200000, 34253542716376896327647232000000, 727956289808441800755158974464000000, 20091593598712993700842387695206400000000

PROG

(MAGMA) [Factorial(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 06 2015

STATUS

proposed

editing

#6 by Michel Marcus at Mon Jul 06 02:55:04 EDT 2015

STATUS

editing

proposed

#5 by Michel Marcus at Mon Jul 06 02:54:31 EDT 2015

COMMENTS

Integral representation as n-th moment of a positive function W(x) expressed in terms of Meijer's G-function on the positive axis, in Maple notation: a(n)= int(x^n*W(x),x=0..infinity)= int(x^n*(1/2)*MeijerG([[], []], [[1, 1/2, 0], []], (1/16)*x) /(sqrt(x)*Pi),x=0..infinity), n=0,1... . Explicit form of the function W(x) is

Explicit form of the function W(x) is

W(x)=1/2*(-1/2*Pi*x^(1/2)*hypergeom([],[1/2, 3/2],-1/16*x)+Pi^(1/2)-1/16*Pi^(1/2)*x*sum((-1)^(2*j)* (-Psi(3/2+j)+Pi*tan(Pi*j)-Psi(2+j)-Psi(1+j)-4*ln(2)+ln(x))*2^(-4*j)* (x^j)*sec(Pi*j)*2^(2*j)/(1/2+j)/GAMMA(1+2*j)/GAMMA(2+j), j = 0..infinity))/(x^(1/2)*Pi);

(x^j)*sec(Pi*j)*2^(2*j)/(1/2+j)/GAMMA(1+2*j)/GAMMA(2+j),

j = 0..infinity))/(x^(1/2)*Pi);

STATUS

proposed

editing

#4 by Jon E. Schoenfield at Sun Jul 05 20:16:00 EDT 2015

STATUS

editing

proposed

#3 by Jon E. Schoenfield at Sun Jul 05 20:15:32 EDT 2015

NAME

a(n) = (2*n)!*(2*n+1)!/n! = n!*A000909(n), n=0,1...

COMMENTS

expressed in terms of Meijer's G-function on the positive axis,

in Maple notation: a(n)= int(x^n*W(x),x=0..infinity)=

intW(x)=1/2*(-1/2*Pi*x^n*(1/2)*MeijerGhypergeom([[], []1/2, 3/2], [[-1, /16*x)+Pi^(1/2, 0], []], ()-1/16*Pi^(1/2)*x*sum((-1)^(2*j)* (-Psi(3/2+j)+Pi*tan(Pi*j)-Psi(2+j)-Psi(1+j)-4*ln(2)+ln(x))*2^(-4*j)*

/(sqrt(x)*Pi),x=0..infinity), n=0,1... . Explicit form of the function W(x) is :

W(x)=1/2*(-1/2*Pi*x^(1/2)*hypergeom([],[1/2, 3/2],-1/16*x)+Pi^(1/2)-1/16*Pi^(1/2)*x*sum((-1)^(2*j)*

(-Psi(3/2+j)+Pi*tan(Pi*j)-Psi(2+j)-Psi(1+j)-4*ln(2)+ln(x))*2^(-4*j)*

FORMULA

Hypergeometric generating function: sum(a(n)*x^n/(n!)^4, n=0..infinity)= -2*EllipticE(4*sqrt(x))/((16*x-1)*Pi).

-2*EllipticE(4*sqrt(x))/((16*x-1)*Pi)

CROSSREFS

Cf. A000909, A000165.

STATUS

approved

editing

Discussion

Sun Jul 05

20:16

Jon E. Schoenfield: (I don't know which of the instances of "ln" to leave alone because of the "in Maple notation" introductions.)