A128652 -id:A128652 - cp's OEIS frontend

A164991 Number of triangular involutions of n. A triangular involution is a square involution with at most three faces.

1, 1, 3, 6, 13, 26, 54, 108, 221, 442, 898, 1796, 3634, 7268, 14668, 29336, 59101, 118202, 237834, 475668, 956198, 1912396, 3841588, 7683176, 15425138, 30850276, 61908564, 123817128, 248377156, 496754312

Offset: 1

Simone Rinaldi (rinaldi(AT)unisi.it), Sep 04 2009

The sequence 2^(n+1) - binomial(n, floor(n/2)), which begins 1,3,6,... has Hankel transform (-1)^n*(2*n+1) (A157142). - Paul Barry, Nov 03 2010

For n >= 2 also row sums of A258445. - Wolfdieter Lang, Jun 27 2015

F. Disanto, A. Frosini, S. Rinaldi, Square Involutions, Proceedings of Permutation Patterns, July, 13-17 2009, Florence.
T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, 19th Conference on Formal Power Series and Algebraic Combinatorics, Tianjin, China, July 2-6, 2007.

F. Disanto, A. Frosini, S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5.
T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO], 2006.

Cf. A128652, A128650, A258445.

Join[{1},Table[2^(n-1)-Binomial[n-2,Floor[(n-2)/2]],{n,2,30}]] (* Harvey P. Dale, Dec 26 2015 *)

a(n) =  2^(n-1) - binomial(n-2, (n-2)\2) \\ Michel Marcus, May 27 2013

a(n) = 2^(n-1) - binomial(n-2, floor((n-2)/2)) for n>1, a(1)=1.
From Wolfdieter Lang, Jun 27 2015: (Start)
a(n) = Sum_{k = 1..2*n-3} A258445(n-1, k), n >= 2.
a(2*k+1) = 4*Sum_{j = 0..(k-2)} binomial(2*k-1,j) + 3*binomial(2*k-1,k-1), k >= 1.
a(2*k) = 4*Sum_{j = 0..(k-2)} binomial(2*(k-1),j) + binomial(2*(k-1),k-1), k >= 1. (End)
(-n+1)*a(n) + 2*(n-1)*a(n-1) + 4*(n-4)*a(n-2) + 8*(-n+4)*a(n-3) = 0. - R. J. Mathar, Aug 09 2017

A164990 Number of square involutions of n.

Original entry on oeis.org

1, 2, 4, 10, 22, 52, 114, 260, 564, 1256, 2698, 5908, 12588, 27224, 57620, 123432, 259816, 552400, 1157466, 2446004, 5105532, 10735352, 22334524, 46766200, 97021272, 202431152, 418935364, 871425160, 1799558584

Offset: 1

Simone Rinaldi (rinaldi(AT)unisi.it), Sep 04 2009

			a(5)=22, in fact the 22 square involutions of 5 are given by all the involutions of 5, which are 26, minus 14325, 15342, 52341, 42315 which are not square.

F. Disanto, A. Frosini, S. Rinaldi, Square Involutions, Proceedings of Permutation Patterns, July 13-17, 2009, Florence.
T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, 19th Conference on Formal Power Series and Algebraic Combinatorics, Tianjin, China, July 2-6, 2007.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Disanto, A. Frosini, S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5.
T. Mansour, S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO], 2006.

Cf. A128652.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( x*(1-x)^2/(1-2*x)^2 - x^3/((1-2*x)*Sqrt(1-4*x^2)) )); // G. C. Greubel, Nov 25 2018

Rest[CoefficientList[Series[x(1-x)^2/(1-2x)^2 - x^3/((1-2x) Sqrt[1-4x^2]), {x, 0, 29}], x]] (* Michael De Vlieger, Nov 25 2018 *)

my(x='x+O('x^30)); Vec(x*(1-x)^2/(1-2*x)^2 - x^3/((1-2*x)*sqrt(1- 4*x^2))) \\ G. C. Greubel, Nov 25 2018

s=(x*(1-x)^2/(1-2*x)^2 -x^3/((1-2*x)*sqrt(1-4*x^2))).series(x, 30);  a= s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 25 2018

a(n) = (n+2)*2^(n-3) - (n-2)*C(n-3,(n-3)/2), n > 1.
G.f.: x*(1-x)^2/(1-2*x)^2 - x^3/((1-2*x)*sqrt(1-4*x^2)).
(n-3)*(n-8)*a(n) + 2*(-n^2 + 10*n - 20)*a(n-1) + 4*(-n^2 + 12*n - 31)*a(n-2) + 8*(n-4)*(n-7)*a(n-3) = 0.- R. J. Mathar, Jul 24 2012

cp's OEIS Frontend

A164991 Number of triangular involutions of n. A triangular involution is a square involution with at most three faces.

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