A027989 - OEIS

%I #29 Nov 01 2024 02:02:58

%S 1,3,10,33,105,324,977,2895,8462,24465,70101,199368,563425,1583643,

%T 4430290,12342849,34262337,94800780,261545777,719697255,1975722326,

%U 5412138033,14796520365,40380240528,110016825025,299285288499

%N a(n) = self-convolution of row n of array T given by A027926.

%C a(n) is the number of all columns in stack polyominoes of perimeter 2n+4. - _Emanuele Munarini_, Apr 07 2011

%H Vincenzo Librandi, <a href="/A027989/b027989.txt">Table of n, a(n) for n = 0..200</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).

%F a(n) = (2/5)*(n + 1)*F(2*n+3) + (1/5)*F(2*n+2) - (4/5)*(n + 1)*F(2*n), where F(n) = A000045(n). - _Ralf Stephan_, May 13 2004

%F From _Emanuele Munarini_, Apr 07 2011: (Start)

%F a(n) = ((4*n + 5)*F(2*n+1) - (2*n + 1)*F(2*n))/5, where F(n) = A000045(n).

%F a(n) = Sum_{k=0..n} binomial(2*n-k, k)*(k + 1).

%F G.f.: (1 - 3*x + 3*x^2)/(1 - 3*x + x^2)^2.

%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). (End)

%t Table[((5+4n)Fibonacci[1+2n]-(1+2n)Fibonacci[2n])/5,{n,0,20}] [_Emanuele Munarini_, Apr 07 2011]

%o (Maxima) makelist(((5+4*n)*fib(1+2*n)-(1+2*n)*fib(2*n))/5,n,0,20); /* _Emanuele Munarini_, Apr 07 2011 */

%o (PARI) Vec((1-3*x+3*x^2)/(1-3*x+x^2)^2+O(x^66)) /* _Joerg Arndt_, Apr 08 2011 */

%Y Cf. A027926, A054142, A172991, A188648.

%K nonn

%O 0,2

%A _Clark Kimberling_