A158616 Table of expansion coefficients [x^m] of the Rayleigh polynomial of index 2n.
1, 1, 2, 11, 5, 38, 14, 946, 1026, 362, 42, 4580, 4324, 1316, 132, 202738, 311387, 185430, 53752, 7640, 429, 3786092, 6425694, 4434158, 1596148, 317136, 33134, 1430, 261868876, 579783114, 547167306, 287834558, 92481350, 18631334, 2305702
Offset: 1
Examples
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 ; Triangle begins: 1, 1, 2, 11,5, 38,14, 946,1026,362,42, 4580,4324,1316,132, 202738,311387,185430,53752,7640,429, ...
Links
- Matthew House, Table of n, a(n) for n = 1..10015 (rows 1..93)
- Nand Kishore, The Rayleigh Polynomial, Proc. AMS 15 (6) (1964) 911-917.
- Nand Kishore, The Rayleigh Function, Proc. AMS 14 (4) (1963) 527-533.
- D. H. Lehmer, Zeros of the Bessel function J_{nu}(x), Math. Comp. 1 (1945), 405-407. Gives first 12 rows.
- D. H. Lehmer, Zeros of the Bessel function J_{nu}(x), Math. Comp., 1 (1943-1945), 405-407. Gives first 12 rows. [Annotated scanned copy]
Programs
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Maple
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: Phi2n := proc(n,nu) local k ; 4^n*mul( (nu+k)^(floor(n/k)),k=1..n)*sig2n(n,nu) ; factor(%) ; end: for n from 1 to 14 do rpoly := Phi2n(n,nu) ; print(coeffs(rpoly)) ; od:
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Mathematica
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; Phi2n[n_, nu_] := 4^n*Product[(nu + k)^Floor[n/k], {k, 1, n}]*sig2n[n, nu]; T[n_] := CoefficientList[Phi2n[n, x], x]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 01 2023, after R. J. Mathar *)