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

%I A158616 #18 Nov 02 2024 03:36:46
%S A158616 1,1,2,11,5,38,14,946,1026,362,42,4580,4324,1316,132,202738,311387,
%T A158616 185430,53752,7640,429,3786092,6425694,4434158,1596148,317136,33134,
%U A158616 1430,261868876,579783114,547167306,287834558,92481350,18631334,2305702
%N A158616 Table of expansion coefficients [x^m] of the Rayleigh polynomial of index 2n.
%H A158616 Matthew House, <a href="/A158616/b158616.txt">Table of n, a(n) for n = 1..10015</a> (rows 1..93)
%H A158616 Nand Kishore, <a href="http://www.jstor.org/stable/2034908">The Rayleigh Polynomial</a>, Proc. AMS 15 (6) (1964) 911-917.
%H A158616 Nand Kishore, <a href="http://www.jstor.org/stable/2034269">The Rayleigh Function</a>, Proc. AMS 14 (4) (1963) 527-533.
%H A158616 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 A158616 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 A158616 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 A158616 Triangle begins:
%e A158616   1,
%e A158616   1,
%e A158616   2,
%e A158616   11,5,
%e A158616   38,14,
%e A158616   946,1026,362,42,
%e A158616   4580,4324,1316,132,
%e A158616   202738,311387,185430,53752,7640,429,
%e A158616   ...
%p A158616 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 A158616 Phi2n := proc(n,nu) local k ; 4^n*mul( (nu+k)^(floor(n/k)),k=1..n)*sig2n(n,nu) ; factor(%) ; end:
%p A158616 for n from 1 to 14 do rpoly := Phi2n(n,nu) ; print(coeffs(rpoly)) ; od:
%t A158616 sig2n[n_, nu_] := sig2n[n, nu] = If[n == 1, 1/4/(nu + 1), Sum[sig2n[k, nu]*sig2n[n - k, nu], {k, 1, n - 1}]/(nu + n)] // Simplify;
%t A158616 Phi2n[n_, nu_] := 4^n*Product[(nu + k)^Floor[n/k], {k, 1, n}]*sig2n[n, nu];
%t A158616 T[n_] := CoefficientList[Phi2n[n, x], x];
%t A158616 Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Dec 01 2023, after _R. J. Mathar_ *)
%Y A158616 Cf. A000992, A000175 (first column), A000331 (2nd column).
%K A158616 nonn,tabf
%O A158616 1,3
%A A158616 _R. J. Mathar_, Mar 22 2009