OFFSET
0,3
REFERENCES
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
LINKS
FORMULA
Recurrence: (n-1)^2*a(n) = (2*n - 1)*(3*n^2 - 3*n + 1)*a(n-1) + n^2*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * 3^(n-1) * n^(n+1) / exp(n-1/3). - Vaclav Kotesovec, Jul 22 2015
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(1/2)_{n}*6^(n - 1)* hypergeometric1f1(1 - n, -2*n, 2/3).
E.g.f.: ((1 - 6*x)^(3/2) + 3*x*(1 - 6*x)^(1/2) + 15*x - 1) * exp((1 - sqrt(1 - 6*x))/3)/(9*(1 - 6*x)^(3/2)). (End)
G.f.: (t/(1-t)^3)*hypergeometric2f0(2,3/2; - ; 6*t/(1-t)^2). - G. C. Greubel, Aug 16 2017
MATHEMATICA
Table[Sum[(n+k+1)!*3^k/((n-k-1)!*k!*2^(k+1)), {k, 0, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0}, Table[2*n*Pochhammer[1/2, n]*6^(n - 1)* Hypergeometric1F1[1 - n, -2*n, 2/3], {n, 1, 50}]] (* G. C. Greubel, Aug 14 2017 *)
PROG
(PARI) for(n=0, 50, print1(sum(k=0, n-1, (n+k+1)!*3^k/((n-k-1)!*k! *2^(k+1))), ", ")) \\ G. C. Greubel, Aug 14 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 08 2001
STATUS
approved