A235136 -id:A235136 - cp's OEIS frontend

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

N. J. A. Sloane, Oct 27 2011

nonn

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

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

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.

Cf. A235136.

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 *)

{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 */

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

A249100 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 21, 12, 3, 1, 45, 48, 21, 3, 1, 231, 177, 81, 32, 3, 1, 585, 855, 450, 120, 45, 3, 1, 3465, 3240, 2070, 930, 165, 60, 3, 1, 9945, 18000, 10890, 4110, 1695, 216, 77, 3, 1, 65835, 71505, 57330, 28560, 7245, 2835, 273, 96, 3, 1, 208845, 443835, 300195, 143640, 64155, 11781, 4452, 336, 117, 3, 1

Offset: 0

Clark Kimberling, Oct 21 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + (2*n+1)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A249101(n) for n >= 0.
(n-th term of column 1) = A235136(n) for n >= 1.

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (3 + x)/1, so that p(1,x) = 3 + x;
f(2,x) = (5 + 3*x + x^2)/(3 + x), so that p(2,x) = 5 + 3*x + x^2.
First 6 rows of the triangle of coefficients:
    1;
    3,   1;
    5,   3,   1;
   21,  12,   3,   1;
   45,  48,  21,   3,   1;
  231, 177,  81,  32,   3,   1;

Clark Kimberling, Rows 0..100, flattened

Cf. A249101, A245136, A087299.

z = 11; p[x_, n_] := x + (2 n - 1)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249100 array *)
Flatten[CoefficientList[u, x]] (* A249100 sequence *)
v = u /. x -> 1  (* A249101 *)
u /. x -> 0  (* A235136 *)
T[ n_Integer, k_Integer] := (T[n, k] = If[n<2, Boole[0==k], T[n-1, k-1] + (2*n-1)*T[n-2 ,k] ]); Join @@ Table[T[n, k], {n, 10}, {k, 0, n-1}] (* Michael Somos, Oct 27 2022 *)

T(n, k) = T(n-1, k-1) + (2*n-1)*T(n-2, k). - Michael Somos, Oct 27 2022

cp's OEIS Frontend

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

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