A000175 Related to zeros of Bessel function. (Formerly M2000 N0790)

1, 1, 2, 11, 38, 946, 4580, 202738, 3786092, 261868876, 1992367192, 2381255244240, 21411255538848, 2902625722978656, 451716954504285504, 319933105641374465472, 3761845343198709705600

Offset

1,3

Comments

Constant term of the Rayleigh polynomials. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010]

References

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).

Links

H. Jamke, Table of n, a(n) for n=1..100. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 03 2010]

M. Delest, J.M. Fedou, Enumeration of skew Ferrers diagrams, preprint LaBRI nA degs 89, Bordeaux, Juin 1989 [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 03 2010]

N. Kishore, The Rayleigh Polynomial, Proc. Amer. Math. Soc. 15, No. 6 (1964), pp. 911-917. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010]

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

Mathematica

pi0[n_] := Product[k^Floor[n/k], {k, 1, n}]; J[v_, m_] := Sum[(-1)^n*(x/2)^( 2*n + v)/(n!*(n+v)!), {n, 0, m}] + O[x]^(2*m+v); p = J[1, 101]/(2*J[0, 101]); Reap[For[n=1, n <= 40, n += 2, Print["a(", (n+1)/2, ") = ", an = SeriesCoefficient[p, n]*pi0[(n+1)/2]*2^(n+1)]; Sow[an]]][[2, 1]] (* Jean-François Alcover, Feb 04 2016, adapted from Herman Jamke's 2nd PARI script *)

Prog

(PARI) alpha(k, n)=if(k<floor(n/2), 2, if(n%2==1, 2, 1))
e(r, k, n)=floor(n/r)-floor(k/r)-floor((n-k)/r)
phi2(n)=if(n<3, return(1), return(sum(k=1, floor(n/2), alpha(k, n)*phi2(k)*phi2(n-k)*prod(r=1, n-1, (v+r)^e(r, k, n)))))
for(m=1, 30, print1(polcoeff(phi2(m), 0)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010
(PARI) pi0(n)=prod(k=1, n, k^floor(n/k))
J(v, m)=sum(n=0, m, (-1)^n*(x/2)^(2*n+v)/(n!*(n+v)!))+O(x^(2*m+v))
p=J(1, 101)/(2*J(0, 101)); forstep(n=1, 200, 2, print((n+1)/2" "polcoeff(p, n)*pi0((n+1)/2)*2^(n+1))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 03 2010

Crossrefs

Sequence in context: A259213 A259658 A350952 * A276659 A187259 A296593

Adjacent sequences: A000172 A000173 A000174 * A000176 A000177 A000178

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010

Status

approved