%I #22 Feb 16 2025 08:33:03

%S 1,1,2,1,6,4,1,12,20,8,1,20,60,56,16,1,30,140,224,144,32,1,42,280,672,

%T 720,352,64,1,56,504,1680,2640,2112,832,128,1,72,840,3696,7920,9152,

%U 5824,1920,256,1,90,1320,7392,20592,32032,29120,15360,4352,512,1,110,1980,13728,48048,96096,116480,87040,39168,9728,1024

%N Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).

%C Sum of terms in row n = A001835(n+1). Sum(k*T(n,k), k=0..n)=A123520(n) (n>=1).

%H G. C. Greubel, <a href="/A123519/b123519.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>

%F T(n,k) = 2^k * binomial(n+k,2*k).

%F G.f.: (1-z)/(1 - 2*z + z^2 - 2*t*z).

%F Sum_{k=0..n} k*T(n,k) = A123520(n) (n>=1).

%F Row polynomials are b(n,2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k) * x^k are the Morgan-Voyce polynomials of A085478. The triangle is made up of the odd-indexed rows of A211956. - _Peter Bala_, May 01 2012

%e T(1,1)=2 because a 2 X 3 grid can be tiled in 2 ways with dominoes so that exactly 2 dominoes are in vertical position: place a horizontal domino above or below two adjacent vertical dominoes.

%p T:=(n,k)->2^k*binomial(n+k,2*k): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%t Table[2^k*Binomial[n + k, 2*k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Oct 14 2017 *)

%t CoefficientList[Table[Sqrt[2] Cosh[(2 n + 1) ArcSinh[Sqrt[x/2]]]/Sqrt[2 + x], {n, 0, 10}] // FunctionExpand // Simplify, x] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)

%t CoefficientList[Table[ChebyshevT[2 n - 1, Sqrt[1 + x/2]]/Sqrt[1 + x/2], {n, 10}], x] (* _Eric W. Weisstein_, Apr 04 2018 *)

%o (PARI) for(n=0,10, for(k=0,n, print1(2^k*binomial(n+k,2*k), ", "))) \\ _G. C. Greubel_, Oct 14 2017

%Y Cf. A001835, A123520. A085478, A211955, A211956.

%K nonn,tabl

%O 0,3

%A _Emeric Deutsch_, Oct 16 2006

%E Terms a(57) onward added by _G. C. Greubel_, Oct 14 2017