A000175 -id:A000175 - cp's OEIS frontend

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

R. J. Mathar, Mar 22 2009

			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,
  ...

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]

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

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:

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 *)

A000331 Related to zeros of Bessel function.

Original entry on oeis.org

5, 14, 1026, 4324, 311387, 6425694, 579783114, 4028104212, 7315072725560, 61358264615344, 9569450876916944, 1632353370882506848, 1365475358484643531856, 15211641461623992544160, 74766806258361827981250240, 936580261005146914634459520, 6083678228249789825160175706880, 1936651082361926268672618636234240, 688115696843061332335070140230720000, 10517068622936239459488783307672335360, 2913914903970372007778735454555848514846720

Offset: 4

N. J. A. Sloane

nonn

a(n) is coefficient of nu in Rayleigh polynomial of index 2n.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matthew House, Table of n, a(n) for n = 4..195
D. H. Lehmer, Zeros of the Bessel function J_{nu}(x), Math. Comp. 1 (1945), 405-407.
D. H. Lehmer, Zeros of the Bessel function J_{nu}(x), Math. Comp., 1 (1943-1945), 405-407. [Annotated scanned copy]
Index entries for sequences related to Bessel functions or polynomials

Cf. 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;
Phi2n[n_, nu_] := 4^n*Product[(nu + k)^Floor[n/k], {k, 1, n}]*sig2n[n, nu];
a[n_] := Coefficient[Phi2n[n, x], x, 1];
Table[a[n], {n, 4, 24}] (* Jean-François Alcover, Dec 01 2023, after R. J. Mathar in A158616 *)

More terms from Sean A. Irvine, Nov 11 2010

cp's OEIS Frontend

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

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