OFFSET
0,3
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
G. C. Greubel, Table of n, a(n) for n = 0..400
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
FORMULA
G.f.: 6*x^2*(1-x)^(-5)*hypergeom([5/2,3],[],2*x/(x-1)^2). - Mark van Hoeij, Nov 07 2011
D-finite with recurrence: (n-2)*(n-1)*a(n) = (2*n - 1)*(n^2 - n + 2)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * n^(n+2) / exp(n-1). - Vaclav Kotesovec, Jul 22 2015
a(n) = n*(n - 1)*(1/2)_{n}*2^n* hypergeometric1F1(2 - n, -2*n, 2), where (a)_{n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017
E.g.f.: (-1)*(1 - 2*x)^(-5/2)*((4 - 14*x + 9*x^2)*sqrt(1 - 2*x) + (2*x^3 - 24*x^2 + 18*x - 4))*exp((1 - sqrt(1 - 2*x))). - G. C. Greubel, Aug 16 2017
MAPLE
(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1, diff(f(n), x$2)), n=0..60)];
MATHEMATICA
Table[Sum[(n+k+2)!/(2^(k+2)*(n-k-2)!*k!), {k, 0, n-2}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[n*(n - 1)*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[2 - n, -2*n, 2], {n, 2, 50}]] (* G. C. Greubel, Aug 14 2017 *)
PROG
(PARI) for(n=0, 20, print1(sum(k=0, n-2, (n+k+2)!/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved