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%I M3065 N1242 #97 Apr 30 2023 04:17:29
%S 1,1,3,19,211,3651,90921,3081513,136407699,7642177651,528579161353,
%T 44237263696473,4405990782649369,515018848029036937,
%U 69818743428262376523,10865441556038181291819,1923889742567310611949459,384565973956329859109177427,86180438505835750284241676121
%N Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.
%C a(n) has the Lucas property, namely a(n) is congruent to a(n_0)a(n_1)...a(n_k) modulo p for any prime p where n_0,n_1,... are the base p digits of n. (Carlitz via McIntosh)
%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 Alois P. Heinz, <a href="/A000275/b000275.txt">Table of n, a(n) for n = 0..261</a>
%H Morton Abramson and David Promislow, <a href="https://doi.org/10.1016/0097-3165(78)90012-2">Enumeration of arrays by column rises</a>, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) (with t=0 and m=2) on p. 249.
%H Leonid Bedratyuk and Nataliia Luno, <a href="https://doi.org/10.15330/cmp.12.1.10-18">Connection problems for the generalized hypergeometric Appell polynomials</a>, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 10-18.
%H L. Carlitz, <a href="http://dx.doi.org/10.1007/BF01900214">The coefficients of the reciprocal of J_0(x)</a>, Archiv. Math. 6 (1955), 121-127.
%H L. Carlitz, Richard Scoville, and Theresa Vaughan, <a href="http://projecteuclid.org/euclid.bams/1183535825">Enumeration of pairs of permutations and sequences</a>, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.
%H L. Carlitz, Richard Scoville, and Theresa Vaughan, <a href="/A259465/a259465.pdf">Enumeration of pairs of permutations and sequences</a>, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. [Annotated scanned copy]
%H L. Carlitz, N. J. A. Sloane, and C. L. Mallows, <a href="/A259465/a259465_1.pdf">Correspondence, 1975</a>.
%H Jan Geuenich, <a href="https://arxiv.org/abs/1803.10707">Tilting modules for the Auslander algebra of K[x]/(xn)</a>, arXiv:1803.10707 [math.RT], 2018.
%H Gunnar Thor Magnússon, <a href="http://arxiv.org/abs/1401.4048">The inner product on exterior powers of a complex vector space</a>, arXiv preprint arXiv:1401.4048 [math.AG], 2014.
%H R. McIntosh, <a href="http://www.jstor.org/stable/2325058">A generalization of a congruential property of Lucas</a>, Amer. Math. Monthly 99(3) (1992), 231-238; see page 232. MR1216210 (95b:11008)
%H J. Riordan, <a href="http://www.jstor.org/stable/2312584">Inverse relations and combinatorial identities</a>, Amer. Math. Monthly, 71 (1964), 485-498.
%H Jonathan D. H. Smith, <a href="http://dx.doi.org/10.1007/BF01393379">Commutative Moufang loops and Bessel functions</a>, Invent. Math. 67(1) (1982), 173-187.
%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F a(n) = Sum_{r=0..n-1} (-1)^(r+n+1) binomial(n, r)^2 a(r), if n > 0.
%F Sum_{n>=0} a(n) * x^n / n!^2 = 1 / J_0(sqrt(4*x)). - _Michael Somos, May 17 2004
%F From _Peter Bala_, Aug 08 2011: (Start)
%F Conjectural formula: 1 = Sum_{n>=0} a(n)*x^n*Sum_{k>=0} binomial(n+k,k)^2*(-x)^k.
%F Apart from the initial term, first column of A192721. (End)
%F E.g.f.: 1/J_0(sqrt(4*x)) = 1 + x/Q(0), where Q(k) = (k+1)^2 - x + (k+1)^2*x/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 06 2013
%F a(n) ~ c * (n!)^2 / r^n, where r = (1/4)*BesselJZero[0,1]^2 = 1.4457964907366961302939989396139517587678604516 and c = 1/(sqrt(r) * BesselJ(1, 2*sqrt(r))) = 1.60197469692804662664846689139151227422675123376219... - _Vaclav Kotesovec_, Mar 02 2014, updated Apr 01 2018
%e From _Peter Bala_, Aug 08 2011: (Start)
%e a(3) = 19: The 19 pairs of permutations in the group S_3 x S_3 with no common rises correspond to the zero entries in the table below.
%e ======================================
%e Number of common rises in S_3 x S_3
%e ======================================
%e | 123 132 213 231 312 321
%e ======================================
%e 123| 2 1 1 1 1 0
%e 132| 1 1 0 1 0 0
%e 213| 1 0 1 0 1 0
%e 231| 1 1 0 1 0 0
%e 312| 1 0 1 0 1 0
%e 321| 0 0 0 0 0 0
%e (End)
%e G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ...
%p A000275 := proc(n) sum(z^k/k!^2, k = 0..infinity);
%p series(%^x, z=0, n+1): n!^2*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
%p seq(A000275(n), n=0..17); # _Peter Luschny_, May 27 2017
%t a[0] = 1; a[n_] := a[n] = Sum[(-1)^(r+n+1)*Binomial[n, r]^2 a[r], {r, 0, n-1}]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Aug 05 2013 *)
%t CoefficientList[Series[1/BesselJ[0,Sqrt[4*x]], {x, 0, 20}], x]* Range[0, 20]!^2 (* _Vaclav Kotesovec_, Mar 02 2014 *)
%t a[ n_] := If[ n < 0, 0, (n! 2^n)^2 SeriesCoefficient[ 1 / BesselJ[ 0, x], {x, 0, 2 n}]]; (* _Michael Somos_, Aug 20 2015 *)
%o (PARI) {a(n) = if( n<0, 0, n!^2 * 4^n * polcoeff( 1 / besselj(0, x + x * O(x^(2*n))), 2*n))}; /* _Michael Somos_, May 17 2004 */
%Y Row 2 of A212855.
%Y Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1), A115368.
%Y Column k=1 of A340986.
%K nonn,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_, Apr 25 2000
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