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 A001518 Bessel polynomial y_n(3). (Formerly M3669 N1495) 18
 1, 4, 37, 559, 11776, 318511, 10522639, 410701432, 18492087079, 943507142461, 53798399207356, 3390242657205889, 233980541746413697, 17551930873638233164, 1421940381306443299981, 123726365104534205331511, 11507973895102987539130504 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77. 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 Gheorghe Coserea and T. D. Noe, Table of n, a(n) for n = 0..200 (terms up to n=100 by T. D. Noe) W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408. Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2009. J. Riordan, Letter to N. J. A. Sloane, Jul. 1968 N. J. A. Sloane, Letter to J. Riordan, Nov. 1970 FORMULA y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!). D-finite with recurrence a(n) = 3(2n-1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006 G.f.: 1/Q(0), where Q(k)= 1 - x - 3*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013 a(n) = exp(1/3)*sqrt(2/(3*Pi))*BesselK(1/2+n,1/3). - Gerry Martens, Jul 22 2015 a(n) ~ sqrt(2) * 6^n * n^n / exp(n-1/3). - Vaclav Kotesovec, Jul 22 2015 E.g.f.: exp(1/3 - 1/3*(1-6*x)^(1/2)) / (1-6*x)^(1/2). (formula due to B. Salvy, see Plouffe link) - Gheorghe Coserea, Aug 06 2015 From G. C. Greubel, Aug 16 2017: (Start) a(n) = (1/2)_{n} * 6^n * hypergeometric1f1(-n; -2*n; 2/3). G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 6*t/(1-t)^2). (End) MAPLE f:= gfun:-rectoproc({a(n)=3*(2*n-1)*a(n-1)+a(n-2), a(0)=1, a(1)=4}, a(n), remember): map(f, [\$0..60]); # Robert Israel, Aug 06 2015 MATHEMATICA Table[Sum[(n+k)!*3^k/(2^k*(n-k)!*k!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *) PROG (PARI) x='x+O('x^33); Vec(serlaplace(exp(1/3 - 1/3 * (1-6*x)^(1/2)) / (1-6*x)^(1/2))) \\ Gheorghe Coserea, Aug 04 2015 CROSSREFS Cf. A001515, A001517. Polynomial coefficients are in A001498. Sequence in context: A316877 A277638 A121080 * A185082 A259822 A036245 Adjacent sequences:  A001515 A001516 A001517 * A001519 A001520 A001521 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified June 2 14:21 EDT 2020. Contains 334787 sequences. (Running on oeis4.)