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A000175 Related to zeros of Bessel function.
(Formerly M2000 N0790)
3
1, 1, 2, 11, 38, 946, 4580, 202738, 3786092, 261868876, 1992367192, 2381255244240, 21411255538848, 2902625722978656, 451716954504285504, 319933105641374465472, 3761845343198709705600 (list; graph; refs; listen; history; text; internal format)
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]
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010
STATUS
approved

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Last modified February 21 07:54 EST 2024. Contains 370219 sequences. (Running on oeis4.)