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 A158616 Table of expansion coefficients [x^m] of the Rayleigh polynomial of index 2n. 2

%I

%S 1,1,2,11,5,38,14,946,1026,362,42,4580,4324,1316,132,202738,311387,

%T 185430,53752,7640,429,3786092,6425694,4434158,1596148,317136,33134,

%U 1430,261868876,579783114,547167306,287834558,92481350,18631334,2305702

%N Table of expansion coefficients [x^m] of the Rayleigh polynomial of index 2n.

%H Nand Kishore, <a href="http://www.jstor.org/stable/2034908">The Rayleigh Polynomial</a>, Proc. AMS 15 (6) (1964) 911-917.

%H Nand Kishore, <a href="http://www.jstor.org/stable/2034269">The Rayleigh Function</a>, Proc. AMS 14 (4) (1963) 527-533.

%H D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-45-99084-3">Zeros of the Bessel function J_{nu}(x)</a>, Math. Comp. 1 (1945), 405-407. Gives first 12 rows.

%H D. H. Lehmer, <a href="/A000175/a000175.pdf">Zeros of the Bessel function J_{nu}(x)</a>, Math. Comp., 1 (1943-1945), 405-407. Gives first 12 rows. [Annotated scanned copy]

%e The polynomials of low index are Phi(2,x)=Phi(4,x) = 1 ; Phi(6,x)=2 ; Phi(8,x)=11+5x ; Phi(10,x)=38+14x ; Phi(12,x)=946+1026x+362x^2+42x^3 ;

%e Triangle begins:

%e 1,

%e 1,

%e 2,

%e 11,5,

%e 38,14,

%e 946,1026,362,42,

%e 4580,4324,1316,132,

%e 202738,311387,185430,53752,7640,429,

%e ...

%p sig2n := proc(n,nu) option remember ; if n = 1 then 1/4/(nu+1) ; else add( procname(k,nu)*procname(n-k,nu),k=1..n-1)/(nu+n) ; simplify(%) ; fi; end:

%p Phi2n := proc(n,nu) local k ; 4^n*mul( (nu+k)^(floor(n/k)),k=1..n)*sig2n(n,nu) ; factor(%) ; end:

%p for n from 1 to 14 do rpoly := Phi2n(n,nu) ; print(coeffs(rpoly)) ; od:

%Y Cf. A000992, A000175 (first column), A000331 (2nd column).

%K nonn,tabf

%O 1,3

%A _R. J. Mathar_, Mar 22 2009

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Last modified July 26 17:37 EDT 2021. Contains 346294 sequences. (Running on oeis4.)