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A120362 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind.

%I A120362 #64 May 04 2017 19:39:51
%S A120362 1,0,-1,0,4,-3,0,-16,60,-45,0,64,-1008,2520,-1575,0,-256,16320,
%T A120362 -105840,189000,-99225,0,1024,-261888,4055040,-15800400,21829500,
%U A120362 -9823275,0,-4096,4193280,-149909760,1153152000,-3178375200,3575672100,-1404728325,0,16384,-67104768,5459650560
%N A120362 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind.
%C A120362 Table has only rows for odd h because all coefficients for even h are zero:
%C A120362 =====|=======================================================================
%C A120362 h \ s|  0     1         2          3            4            5             6
%C A120362 -----|-----------------------------------------------------------------------
%C A120362 1    |  1
%C A120362 3    |  0    -1
%C A120362 5    |  0     4        -3
%C A120362 7    |  0   -16        60        -45
%C A120362 9    |  0    64     -1008       2520        -1575
%C A120362 11   |  0  -256     16320    -105840       189000       -99225
%C A120362 13   |  0  1024   -261888    4055040    -15800400     21829500      -9823275
%C A120362 15   |  0 -4096   4193280 -149909760   1153152000  -3178375200    3575672100
%C A120362 17   |  0 16384 -67104768 5459650560 -79048569600 390486096000 -829555927200
%C A120362 ...
%C A120362 From _Francesco Franco_, Jan 12 2016: (Start)
%C A120362 Conjecture:
%C A120362 If t(h,s) is any term of the previous table after the first column (s>0), then:
%C A120362 t(h,s) = -( 4*s^2*t(h-2,s) + Sum_{j=0..s-1} (t(h-2,j) + t(h,j)) ), with t(1,0) = 1, t(h,0) = 0 for h>1 and t(h,s) = 0 for odd h = 1..2*s-1.
%C A120362 Version without the summation:
%C A120362 t(h,s) = -( 4*s^2*t(h-2,s) - (4*(s-1)^2-1)*t(h-2,s-1) ).
%C A120362 Some example (starting from j=1 in the summation):
%C A120362 t(11,3) = -( 4*t(9,3)*3^2 + Sum_{j=1..2} (t(9,j) + t(11,j)) ) = -( 4*2520*9 + (64-256) + (-1008+16320) ) = -105840; second version:
%C A120362 t(17,5) = -( 4*5^2*t(15,5) - (4*4^2-1)*t(15,4) ) = -( 4*25*(-3178375200) - 63*1153152000 ) = 390486096000.
%C A120362 Also:
%C A120362 t(h,1) = (-1)^(h/2-1/2)*A000302(h/2-3/2) for h>1;
%C A120362 t(h,2) = (-1)^(h/2-3/2)*A115490(h/2-3/2) for h>3;
%C A120362 a(A000124(n)) = 0.
%C A120362 (End)
%H A120362 R. J. Mathar, <a href="/A120362/a120362.pdf">Chebyshev series expansion of the Elliptic Integral of the Second Kind</a>
%F A120362 E(m,phi) = Int_{theta=0..phi} sqrt(1-m*sin^2 theta) d theta.
%F A120362 E(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 A120362 E(m,phi) = phi - m*phi^3/3! + (4*m-3*m^2)*phi^5/5! + (-16*m+60*m^2-45*m^3)*phi^7/7! + ...
%e A120362 so the first row (order phi^1) is a(1,1)=1 for the coefficient of phi,
%e A120362 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 A120362 an := proc(m,n,s) local f: f := coeftayl(EllipticE(sin(phi),m^(1/2)),phi=0,n); coeftayl(f*n!,m=0,s) ; end: nmax := 27 ; 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 A120362 a[n_, s_] := SeriesCoefficient[EllipticE[phi, m], {phi, 0, n}, {m, 0, s}]*n!; Table[a[n, s], {n, 1, 17, 2}, {s, 0, n/2}] // Flatten (* _Jean-François Alcover_, Jan 06 2014 *)
%o A120362 (PARI) {T(n, k) = my(m = 2*n+1); if( k<0 || n<k, 0, m! * polcoeff( polcoeff( intformal( sqrt( 1 - y * sin(x + x * O(x^m))^2 ) ), m), k))}; /* _Michael Somos_, May 04 2017 */
%Y A120362 Cf. A010370, A079484 (diagonal).
%K A120362 sign,tabl,easy
%O A120362 1,5
%A A120362 _R. J. Mathar_, Jun 26 2006