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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.
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
Polynomial coefficients are in A001498.
Sequence in context: A352237 A349714 A121080 * A185082 A259822 A345102
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)