OFFSET
1,3
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]
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
KEYWORD
nonn,tabf
AUTHOR
R. J. Mathar, Mar 22 2009
STATUS
approved