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

Author

Francesco Franco, Apr 23 2016

Keywords

• sign
• tabl

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 9

7 | 0 16 -180 225

9 | 0 -64 3024 -12600 11025

11 | 0 256 -48960 529200 -1323000 893025

13 | 0 -1024 785664 -20275200 110602800 -196465500 108056025

15 | 0 4096 -12579840 749548800 -8072064000 28605376800 39332393100

17 | 0 -16384 201314304 -27298252800 553339987200 -3514374864000 9125115199200

...

Conjecture:

If t(h,s) is any term of the previous table after the first column (s>0), then:

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:

t(11,3) = -((2*3)^2*t(9,5) - (2*3-1)^2*t(9,2)) = -(36*(-12600) - 25*3024) = 529200;

t(17,5) = -((2*5)^2*t(15,5) - (2*5-1)^2*t(15,4)) = -(100*(28605376800) - 81*(-8072064000)) = -351437486400.

Also:

t(h,1) = (-1)^(h/2+1/2)*A000302(h/2-3/2) for h>1;

t(h,2) = (-1)^(h/2-1/2)*(16*t(h-2,2)+9*2^(h-5)) for h>3.

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