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A272102 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the first kind.

%I A272102 #37 Apr 29 2016 17:06:59
%S A272102 1,0,1,0,-4,9,0,16,-180,225,0,-64,3024,-12600,11025,0,256,-48960,
%T A272102 529200,-1323000,893025,0,-1024,785664,-20275200,110602800,-196465500,
%U A272102 108056025,0,4096,-12579840,749548800,-8072064000,28605376800,-39332393100,18261468225
%N A272102 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the first kind.
%C A272102 Table has only rows for odd h because all coefficients for even h are zero:
%C A272102 ===|==========================================================================
%C A272102 h\s| 0      1         2            3            4              5             6
%C A272102 ---|--------------------------------------------------------------------------
%C A272102 1  | 1
%C A272102 3  | 0      1
%C A272102 5  | 0     -4         9
%C A272102 7  | 0     16      -180          225
%C A272102 9  | 0    -64      3024       -12600        11025
%C A272102 11 | 0    256    -48960       529200     -1323000         893025
%C A272102 13 | 0  -1024    785664    -20275200    110602800     -196465500     108056025
%C A272102 15 | 0   4096 -12579840    749548800  -8072064000    28605376800   39332393100
%C A272102 17 | 0 -16384 201314304 -27298252800 553339987200 -3514374864000 9125115199200
%C A272102 ...
%C A272102 Conjecture:
%C A272102 If t(h,s) is any term of the previous table after the first column (s>0), then:
%C A272102 t(h,s) = -( (2*s)^2*t(h-2,s) - (2*s-1)^2*t(h-2,s-1) ), with t(1,0) = 1, t(h,0) = 0 for h>1 and t(h,s) = 0 for odd h = 1..2*s-1. Some example:
%C A272102 t(11,3) = -((2*3)^2*t(9,5) - (2*3-1)^2*t(9,2)) = -(36*(-12600) - 25*3024) = 529200;
%C A272102 t(17,5) = -((2*5)^2*t(15,5) - (2*5-1)^2*t(15,4)) = -(100*(28605376800) - 81*(-8072064000)) = -351437486400.
%C A272102 Also:
%C A272102 t(h,1) = (-1)^(h/2+1/2)*A000302(h/2-3/2) for h>1;
%C A272102 t(h,2) = (-1)^(h/2-1/2)*(16*t(h-2,2)+9*2^(h-5)) for h>3.
%H A272102 Francesco Franco, <a href="/A272102/b272102.txt">Rows n = 1..14 of triangle, flattened</a>
%F A272102 F(m,phi) = Int_{theta=0..phi} 1/sqrt(1-m*sin^2 theta) d theta.
%F A272102 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!.
%e A272102 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! + ...
%e A272102 so the first row (order phi^1) is a(1,1)=1 for the coefficient of phi,
%e A272102 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!.
%p A272102 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;
%t A272102 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
%Y A272102 Cf. A120362.
%K A272102 sign,tabl
%O A272102 1,5
%A A272102 _Francesco Franco_, Apr 23 2016
%E A272102 Missing 0 added by _Bruno Berselli_, Apr 29 2016