A272102 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the first kind.
1, 0, 1, 0, -4, 9, 0, 16, -180, 225, 0, -64, 3024, -12600, 11025, 0, 256, -48960, 529200, -1323000, 893025, 0, -1024, 785664, -20275200, 110602800, -196465500, 108056025, 0, 4096, -12579840, 749548800, -8072064000, 28605376800, -39332393100, 18261468225
Offset: 1
Examples
F(m,phi) = phi + m*phi^3/3! + (-4*m+9*m^2)*phi^5/5! + (16*m-180*m^2+225*m^3)*phi^7/7! + (-64*m+3024*m^2-12600*m^3+11025*m^4)*phi^9/9! + ... so the first row (order phi^1) is a(1,1)=1 for the coefficient of phi, the second row (order phi^3) is a(2,0)=0 for the missing coefficient of m^0*phi^3, and a(2,1)=-1 for the coefficient of m^1*phi^3/3!.
Links
- Francesco Franco, Rows n = 1..14 of triangle, flattened
Crossrefs
Cf. A120362.
Programs
-
Maple
an := proc(m, n, s) local f: f := coeftayl(EllipticF(sin(phi), m^(1/2)), phi=0, n); coeftayl(f*n!, m=0, s) ; end: nmax := 28 ; for n from 1 to nmax by 2 do for s from 0 to (n-1)/2 do printf("%d, ", an(m, n, s)) ; od ; od; -
Mathematica
a[1, 0] = 1; a[n_, s_] := SeriesCoefficient[EllipticF[phi, m], {phi, 0, n}, {m, 0, s}]*n!; Table[a[n, s], {n, 1, 17, 2}, {s, 0, n/2}] // Flatten
Formula
F(m,phi) = Int_{theta=0..phi} 1/sqrt(1-m*sin^2 theta) d theta.
F(m,phi) = Sum_{n=1,3,5,7,9,...} ( Sum_{s=0..(n-1)/2} a((n-1)/2,s) * m^s)*phi^n/n!.
Extensions
Missing 0 added by Bruno Berselli, Apr 29 2016
Comments