cp's OEIS Frontend

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

%I A196265 #36 Jul 25 2020 18:53:43
%S A196265 1,2,4,8,26,66,276,816,4050,13410,75780,274680,1723050,6735330,
%T A196265 46104660,192296160,1418802210,6264006210,49355252100,229233450600,
%U A196265 1914861598650,9309854203650,81969299111700,415483465597200,3837397323409650,20209910950879650
%N A196265 Number of standard puzzles of shape 2 X n with support CK (see reference for precise definition).
%C A196265 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.
%C A196265 The sequence appears on pages 4 and 13 of the Han reference. a(1)=1 by convention. - _Michael Somos_, Jan 16 2014
%H A196265 Alois P. Heinz, <a href="/A196265/b196265.txt">Table of n, a(n) for n = 1..500</a>
%H A196265 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A196265 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%F A196265 a(n) = A235136(n-1) + A235136(n-2) if n > 1. - _Michael Somos_, Jan 16 2014
%F A196265 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
%e A196265 G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 26*x^5 + 66*x^6 + 276*x^7 + 816*x^8 + ...
%t A196265 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 *)
%o A196265 (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 */
%Y A196265 Cf. A235136.
%K A196265 nonn
%O A196265 1,2
%A A196265 _N. J. A. Sloane_, Oct 27 2011