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A000175 Related to zeros of Bessel function.
(Formerly M2000 N0790)
3

%I M2000 N0790 #29 Aug 23 2016 04:09:31

%S 1,1,2,11,38,946,4580,202738,3786092,261868876,1992367192,

%T 2381255244240,21411255538848,2902625722978656,451716954504285504,

%U 319933105641374465472,3761845343198709705600

%N Related to zeros of Bessel function.

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H H. Jamke, <a href="/A000175/b000175.txt">Table of n, a(n) for n=1..100</a>. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 03 2010]

%H M. Delest, J.M. Fedou, <a href="http://www.mathe2.uni-bayreuth.de/axel/papers/fedou:enumeration_of_skew_ferrers_diagrams.ps.gz">Enumeration of skew Ferrers diagrams</a>, preprint LaBRI nA degs 89, Bordeaux, Juin 1989 [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 03 2010]

%H N. Kishore, <a href="http://www.ams.org/jams/2009-22-02/S0894-0347-08-00618-8/S0894-0347-08-00618-8.pdf">The Rayleigh Polynomial</a>, Proc. Amer. Math. Soc. 15, No. 6 (1964), pp. 911-917. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010]

%H D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-45-99084-3">Zeros of the Bessel function J_{nu}(x)</a>, Math. Comp. 1 (1945), 405-407.

%H D. H. Lehmer, <a href="/A000175/a000175.pdf">Zeros of the Bessel function J_{nu}(x)</a>, Math. Comp., 1 (1943-1945), 405-407. [Annotated scanned copy]

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%t 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 *)

%o (PARI) alpha(k,n)=if(k<floor(n/2),2,if(n%2==1,2,1))

%o e(r,k,n)=floor(n/r)-floor(k/r)-floor((n-k)/r)

%o 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)))))

%o for(m=1,30,print1(polcoeff(phi2(m),0)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Aug 20 2010

%o (PARI) pi0(n)=prod(k=1,n,k^floor(n/k))

%o J(v,m)=sum(n=0,m,(-1)^n*(x/2)^(2*n+v)/(n!*(n+v)!))+O(x^(2*m+v))

%o 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

%K nonn

%O 1,3

%A _N. J. A. Sloane_

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

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)