OFFSET
0,3
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
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 16 2006
EXTENSIONS
Terms a(57) onward added by G. C. Greubel, Oct 14 2017
STATUS
approved