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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved