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

1,3

Links

Table of n, a(n) for n=1..35.

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]

Example

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,
  ...

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:

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 *)

Crossrefs

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

Sequence in context: A087552 A124688 A339807 * A127821 A114724 A226219

Adjacent sequences: A158613 A158614 A158615 * A158617 A158618 A158619

Keyword

nonn,tabf

Author

R. J. Mathar, Mar 22 2009

Status

approved