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%I #37 Dec 28 2015 15:52:07
%S 1,2,5,15,51,189,748,3128,13731,62969,300552,1488704,7634723,40464741,
%T 221311617,1247444859,7238458309,43196661875,264878725516,
%U 1667564565616,10770316016557,71314258947903,483765644021787,3359905164274725
%N Binomial transform of Bessel numbers A006789.
%C Prefaced with a 1: (1, 1, 2, 5, 15, 51, ...), convolved with A006789 = A006789; identical to reversing k terms of one sequence then taking the dot product of k terms of the other: e.g., A006789(6) = 143 = (51, 15, 5, 2, 1, 1) dot (1, 1, 2, 5, 14, 43) = (51 + 15 + 10 + 10 + 14 + 43). Equals row sums of triangle A153199. A153197 can be generated from the Hankel transform [1,1,1,...] by taking successive iterates of the operations: (binomial transform of [1,1,1,...]) followed by INVERT transform of the result, then binomial transform of the result, (repeat cycle)...; until the operations converge upon a two sequence fixed limit cycle of A006789 and A153197. Or, the infinite set of operations Q may begin: INVERT transform of [1,1,1,..] followed by binomial transform of the result, INVERT transform of the result, etc; until the operations again converge upon A006789 and A153197. The two sequences A006789 and A153197 have the mutual relationships that binomial transform of A006789 = A153197; while the INVERT transform of A153197 prefaced with a 1 and then prefaced with a 1 afterward = A006789.
%C Product of the two sequences (A153197 prefaced with a 1) and A006789 = A006789 with offset 1. Or, (1,1,2,5,15,51,...) * (1,1,2,5,14,43,...) = (1,2,5,14,43,...).
%C Conjecture: Given any sequence with Hankel transform of [1,1,1,...], performing alternate operations: binomial transform followed by INVERT transform, then binomial transform of the last result (repeat); or INVERT transform starting first, will converge upon A006789 and A153197 as a two sequence limit cycle. The conjecture can be extended to any Hankel transform (and their accompanying sequence set): analogous operations will converge upon a Bessel-type sequence and its binomial transform.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%F INVERT transform of A153197 with 1 prepended is A006789 with initial 1 removed. Bessel numbers with offset 1: (1, 2, 5, 14, 43, 143, ...).
%F If the offset is 1, the g.f. A(x) satisfies A(x) = x / (1 - 2*x - x * A( x / (1 - x))). - _Michael Somos_, Mar 06 2011
%F G.f.: 1/U(0) where U(k)= 1 - x*(k+2) - x^2/U(k+1) ; (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 11 2012
%F G.f.: 1/x^2-1/x-U(0)/x^2 where U(k)= 1 - x*(k+1) - x^2/U(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 24 2012
%e a(3) = 15 = (1, 3, 3, 1) dot (1, 1, 2, 5) = (1 + 3 + 6 + 5).
%e G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 51*x^4 + 189*x^5 + 748*x^6 + 3128*x^7 + ...
%t a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Nest[ Series[ x / (1 - 2 x - x (Normal[#] /. x -> x / (1 - x))), {x, 0, m}] &, 0, Ceiling[m/2]], {x, 0, m}]]]; (* _Michael Somos_, Aug 05 2014 *)
%o (PARI) {a(n) = local( A = O(x) ); if( n<0, 0, n++; for( k=1, ceil(n/2), A = x / (1 - 2*x - x * subst( A, x, x / (1 - x)))); polcoeff( A, n))}; /* _Michael Somos_, Mar 06 2011 */
%Y Cf. A153198, A153199, A006789.
%K nonn
%O 0,2
%A _Gary W. Adamson_, Dec 20 2008
%E Edited by _N. J. A. Sloane_, Mar 06 2011
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