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 A000275 Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden. (Formerly M3065 N1242) 28
 1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427, 86180438505835750284241676121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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) 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 Alois P. Heinz, Table of n, a(n) for n = 0..261 Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) (with t=0 and m=2) on p. 249. Leonid Bedratyuk and Nataliia Luno, Connection problems for the generalized hypergeometric Appell polynomials, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 10-18. L. Carlitz, The coefficients of the reciprocal of J_0(x), Archiv. Math. 6 (1955), 121-127. L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. [Annotated scanned copy] L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975. Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(xn), arXiv:1803.10707 [math.RT], 2018. Gunnar Thor Magnússon, The inner product on exterior powers of a complex vector space, arXiv preprint arXiv:1401.4048 [math.AG], 2014. R. McIntosh, A generalization of a congruential property of Lucas, Amer. Math. Monthly 99(3) (1992), 231-238; see page 232. MR1216210 (95b:11008) J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498. Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67(1) (1982), 173-187. Index entries for sequences related to Bessel functions or polynomials FORMULA a(n) = Sum_{r=0..n-1} (-1)^(r+n+1) binomial(n, r)^2 a(r), if n > 0. Sum_{n>=0} a(n) * x^n / n!^2 = 1 / J_0(sqrt(4*x)). - _Michael Somos, May 17 2004 From Peter Bala, Aug 08 2011: (Start) Conjectural formula: 1 = Sum_{n>=0} a(n)*x^n*Sum_{k>=0} binomial(n+k,k)^2*(-x)^k. Apart from the initial term, first column of A192721. (End) 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 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 EXAMPLE From Peter Bala, Aug 08 2011: (Start) 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. ====================================== Number of common rises in S_3 x S_3 ====================================== | 123 132 213 231 312 321 ====================================== 123| 2 1 1 1 1 0 132| 1 1 0 1 0 0 213| 1 0 1 0 1 0 231| 1 1 0 1 0 0 312| 1 0 1 0 1 0 321| 0 0 0 0 0 0 (End) G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ... MAPLE A000275 := proc(n) sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1): n!^2*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end: seq(A000275(n), n=0..17); # Peter Luschny, May 27 2017 MATHEMATICA 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 *) CoefficientList[Series[1/BesselJ[0, Sqrt[4*x]], {x, 0, 20}], x]* Range[0, 20]!^2 (* Vaclav Kotesovec, Mar 02 2014 *) a[ n_] := If[ n < 0, 0, (n! 2^n)^2 SeriesCoefficient[ 1 / BesselJ[ 0, x], {x, 0, 2 n}]]; (* Michael Somos, Aug 20 2015 *) PROG (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 */ CROSSREFS Row 2 of A212855. Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1), A115368. Column k=1 of A340986. Sequence in context: A049056 A204262 A165356 * A058165 A074707 A230317 Adjacent sequences: A000272 A000273 A000274 * A000276 A000277 A000278 KEYWORD nonn,nice AUTHOR N. J. A. Sloane EXTENSIONS More terms from Christian G. Bower, Apr 25 2000 STATUS approved

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Last modified September 25 15:05 EDT 2023. Contains 365648 sequences. (Running on oeis4.)