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A001518
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Bessel polynomial y_n(3).
(Formerly M3669 N1495)
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18
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1, 4, 37, 559, 11776, 318511, 10522639, 410701432, 18492087079, 943507142461, 53798399207356, 3390242657205889, 233980541746413697, 17551930873638233164, 1421940381306443299981, 123726365104534205331511, 11507973895102987539130504
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OFFSET
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0,2
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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
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)
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MAPLE
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f:= gfun:-rectoproc({a(n)=3*(2*n-1)*a(n-1)+a(n-2), a(0)=1, a(1)=4}, a(n), remember):
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MATHEMATICA
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Table[Sum[(n+k)!*3^k/(2^k*(n-k)!*k!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)
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PROG
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(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
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CROSSREFS
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Polynomial coefficients are in A001498.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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