A123519 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).

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, 720, 352, 64, 1, 56, 504, 1680, 2640, 2112, 832, 128, 1, 72, 840, 3696, 7920, 9152, 5824, 1920, 256, 1, 90, 1320, 7392, 20592, 32032, 29120, 15360, 4352, 512, 1, 110, 1980, 13728, 48048, 96096, 116480, 87040, 39168, 9728, 1024

Offset

0,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Formula

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

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

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

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

Example

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.

Maple

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

Mathematica

Table[2^k*Binomial[n + k, 2*k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 14 2017 *)
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 *)
CoefficientList[Table[ChebyshevT[2 n - 1, Sqrt[1 + x/2]]/Sqrt[1 + x/2], {n, 10}], x] (* Eric W. Weisstein, Apr 04 2018 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(2^k*binomial(n+k, 2*k), ", "))) \\ G. C. Greubel, Oct 14 2017

Crossrefs

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

Sequence in context: A208913 A208911 A208761 * A167024 A114687 A137594

Adjacent sequences: A123516 A123517 A123518 * A123520 A123521 A123522

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 16 2006

Extensions

Terms a(57) onward added by G. C. Greubel, Oct 14 2017

Status

approved