OFFSET
1,2
COMMENTS
The Han reference contains many sequences not yet in the OEIS (as well as over 100 that are). This is the first one that was not already in the OEIS.
The sequence appears on pages 4 and 13 of the Han reference. a(1)=1 by convention. - Michael Somos, Jan 16 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..500
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
FORMULA
E.g.f. A(x) =: y satisfies 0 = -(1 + x)^2 + y * x - y' * (1 + 2*x + 2*x^2) + y'' * (1 + x) = (1 - x) + y' * (1 - x) - y'' * (1 + 2*x^2) + y''' * x. - Michael Somos, Jan 16 2014
EXAMPLE
G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 26*x^5 + 66*x^6 + 276*x^7 + 816*x^8 + ...
MATHEMATICA
a[ n_] := If[ n < 2, Boole[n == 1], With[{m = Mod[n, 2]}, 2^(n - m) (Pochhammer[ 1/4 + m/2, (n - m)/2] - (-1)^ m Pochhammer[ -1/4 + m/2, (n - m)/2]) ]]; (* Michael Somos, Jan 16 2014 *)
PROG
(PARI) {a(n) = my(v=[1, 1]); if( n<2, n==1, for(k=1, n-1, v = [v[2], v[1] * (2*k-1)]); v[1] + v[2])}; /* Michael Somos, Jan 16 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 27 2011
STATUS
approved