A247497 - OEIS

%I #11 Dec 14 2014 08:08:29

%S 1,1,1,2,3,2,4,10,12,6,9,33,62,60,24,21,111,300,450,360,120,51,378,

%T 1412,3000,3720,2520,720,127,1303,6552,18816,32760,34440,20160,5040,

%U 323,4539,30186,113820,264264,388080,352800,181440,40320

%N 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).

%F 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:

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

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

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

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

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

%e Triangle starts:

%e [  1],

%e [  1,    1],

%e [  2,    3,    2],

%e [  4,   10,   12,     6],

%e [  9,   33,   62,    60,    24],

%e [ 21,  111,  300,   450,   360,   120],

%e [ 51,  378, 1412,  3000,  3720,  2520,   720],

%e [127, 1303, 6552, 18816, 32760, 34440, 20160, 5040].

%e .

%e [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,...

%e [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,...

%p A247497_row := proc(n) local A, M, p;

%p A := (n,k) -> `if`(type(n-k, odd),0,n!/(k!*((n-k)/2)!^2*((n-k)/2+1))):

%p M := (k,x) -> add(A(k,j)*x^j,j=0..k): # Motzkin polynomial

%p p := expand(sum(x^k*M(n,k),k=0..infinity));

%p [seq((-1)^(n+1)*coeff(convert(p,parfrac),(x-1)^(-j)),j=1..n+1)] end:

%p seq(print(A247497_row(n)),n=0..7);

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

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Dec 13 2014