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A120362
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Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind.
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2
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1, 0, -1, 0, 4, -3, 0, -16, 60, -45, 0, 64, -1008, 2520, -1575, 0, -256, 16320, -105840, 189000, -99225, 0, 1024, -261888, 4055040, -15800400, 21829500, -9823275, 0, -4096, 4193280, -149909760, 1153152000, -3178375200, 3575672100, -1404728325, 0, 16384, -67104768, 5459650560
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OFFSET
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1,5
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COMMENTS
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Table has only rows for odd h because all coefficients for even h are zero:
=====|=======================================================================
h \ s| 0 1 2 3 4 5 6
-----|-----------------------------------------------------------------------
1 | 1
3 | 0 -1
5 | 0 4 -3
7 | 0 -16 60 -45
9 | 0 64 -1008 2520 -1575
11 | 0 -256 16320 -105840 189000 -99225
13 | 0 1024 -261888 4055040 -15800400 21829500 -9823275
15 | 0 -4096 4193280 -149909760 1153152000 -3178375200 3575672100
17 | 0 16384 -67104768 5459650560 -79048569600 390486096000 -829555927200
...
Conjecture:
If t(h,s) is any term of the previous table after the first column (s>0), then:
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.
Version without the summation:
t(h,s) = -( 4*s^2*t(h-2,s) - (4*(s-1)^2-1)*t(h-2,s-1) ).
Some example (starting from j=1 in the summation):
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:
t(17,5) = -( 4*5^2*t(15,5) - (4*4^2-1)*t(15,4) ) = -( 4*25*(-3178375200) - 63*1153152000 ) = 390486096000.
Also:
t(h,1) = (-1)^(h/2-1/2)*A000302(h/2-3/2) for h>1;
t(h,2) = (-1)^(h/2-3/2)*A115490(h/2-3/2) for h>3;
(End)
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LINKS
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FORMULA
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E(m,phi) = Int_{theta=0..phi} sqrt(1-m*sin^2 theta) d theta.
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!.
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EXAMPLE
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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! + ...
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!.
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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