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 A120362 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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 ... From Francesco Franco, Jan 12 2016: (Start) 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; a(A000124(n)) = 0. (End) LINKS Table of n, a(n) for n=1..40. R. J. Mathar, Chebyshev series expansion of the Elliptic Integral of the Second Kind FORMULA 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!. EXAMPLE 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!. MAPLE 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; MATHEMATICA 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 *) PROG (PARI) {T(n, k) = my(m = 2*n+1); if( k<0 || n

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Last modified September 27 20:41 EDT 2023. Contains 365714 sequences. (Running on oeis4.)