A247497 Triangle read by rows, T(n,k) (n>=0, 0<=k<=n) coefficients of the partial fraction decomposition of rational functions generating the columns of A247495 (the Motzkin polynomials evaluated at nonnegative integers).

1, 1, 1, 2, 3, 2, 4, 10, 12, 6, 9, 33, 62, 60, 24, 21, 111, 300, 450, 360, 120, 51, 378, 1412, 3000, 3720, 2520, 720, 127, 1303, 6552, 18816, 32760, 34440, 20160, 5040, 323, 4539, 30186, 113820, 264264, 388080, 352800, 181440, 40320

Offset: 0

Author

Peter Luschny, Dec 13 2014

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[  1],
[  1,    1],
[  2,    3,    2],
[  4,   10,   12,     6],
[  9,   33,   62,    60,    24],
[ 21,  111,  300,   450,   360,   120],
[ 51,  378, 1412,  3000,  3720,  2520,   720],
[127, 1303, 6552, 18816, 32760, 34440, 20160, 5040].
.
[n=3] -> [4,10,12,6] -> 4/(x-1)+10/(x-1)^2+12/(x-1)^3+6/(x-1)^4 = 2*x*(-x+2*x^2+2)/(x-1)^4; generating function of A247495[n,3] = 0,4,14, 36,...
[n=4] -> [9,33,62,60,24] -> -9/(x-1)-33/(x-1)^2-62/(x-1)^3-60/(x-1)^4-24/(x-1)^5 = -(2-x-3*x^3+17*x^2+9*x^4)/(x-1)^5; generating function of A247495[n,4] = 2,9,42,137,...
		

Crossrefs

Cf. A247495, A097610, A001006, A058987, A001710.

Programs

Maple

A247497_row := proc(n) local A, M, p;
A := (n,k) -> `if`(type(n-k, odd),0,n!/(k!*((n-k)/2)!^2*((n-k)/2+1))):
M := (k,x) -> add(A(k,j)*x^j,j=0..k): # Motzkin polynomial
p := expand(sum(x^k*M(n,k),k=0..infinity));
[seq((-1)^(n+1)*coeff(convert(p,parfrac),(x-1)^(-j)),j=1..n+1)] end:
seq(print(A247497_row(n)),n=0..7);

Formula

Let M_{n}(x) = sum_{k=0..n} A097610(n,k)*x^k denote the Motzkin polynomials. The T(n,k) are implicitly defined by:

sum_{k=0..n} (-1)^(n+1)*T(n,k)/(x-1)^(k+1) = sum_{k>=0} x^k*M_n(k).

T(n, 0) = A001006(n) (Motzkin numbers).

T(n, n) = A000142(n) = n!.

T(n, 1) = A058987(n+1) for n>=1.

T(n,n-1)= A001710(n+1) for n>=1.