A196265 Number of standard puzzles of shape 2 X n with support CK (see reference for precise definition).

1, 2, 4, 8, 26, 66, 276, 816, 4050, 13410, 75780, 274680, 1723050, 6735330, 46104660, 192296160, 1418802210, 6264006210, 49355252100, 229233450600, 1914861598650, 9309854203650, 81969299111700, 415483465597200, 3837397323409650, 20209910950879650

Offset: 1

Author

N. J. A. Sloane, Oct 27 2011

Keywords

• nonn

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

Examples

			G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 26*x^5 + 66*x^6 + 276*x^7 + 816*x^8 + ...
		

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.

Crossrefs

Cf. A235136.

Programs

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 *)

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 */

Formula

a(n) = A235136(n-1) + A235136(n-2) if n > 1. - Michael Somos, Jan 16 2014

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