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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).

%I A247497 #11 Dec 14 2014 08:08:29
%S A247497 1,1,1,2,3,2,4,10,12,6,9,33,62,60,24,21,111,300,450,360,120,51,378,
%T A247497 1412,3000,3720,2520,720,127,1303,6552,18816,32760,34440,20160,5040,
%U A247497 323,4539,30186,113820,264264,388080,352800,181440,40320
%N 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).
%F A247497 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 A247497 sum_{k=0..n} (-1)^(n+1)*T(n,k)/(x-1)^(k+1) = sum_{k>=0} x^k*M_n(k).
%F A247497 T(n, 0) = A001006(n) (Motzkin numbers).
%F A247497 T(n, n) = A000142(n) = n!.
%F A247497 T(n, 1) = A058987(n+1) for n>=1.
%F A247497 T(n,n-1)= A001710(n+1) for n>=1.
%e A247497 Triangle starts:
%e A247497 [  1],
%e A247497 [  1,    1],
%e A247497 [  2,    3,    2],
%e A247497 [  4,   10,   12,     6],
%e A247497 [  9,   33,   62,    60,    24],
%e A247497 [ 21,  111,  300,   450,   360,   120],
%e A247497 [ 51,  378, 1412,  3000,  3720,  2520,   720],
%e A247497 [127, 1303, 6552, 18816, 32760, 34440, 20160, 5040].
%e A247497 .
%e A247497 [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 A247497 [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 A247497_row := proc(n) local A, M, p;
%p A247497 A := (n,k) -> `if`(type(n-k, odd),0,n!/(k!*((n-k)/2)!^2*((n-k)/2+1))):
%p A247497 M := (k,x) -> add(A(k,j)*x^j,j=0..k): # Motzkin polynomial
%p A247497 p := expand(sum(x^k*M(n,k),k=0..infinity));
%p A247497 [seq((-1)^(n+1)*coeff(convert(p,parfrac),(x-1)^(-j)),j=1..n+1)] end:
%p A247497 seq(print(A247497_row(n)),n=0..7);
%Y A247497 Cf. A247495, A097610, A001006, A058987, A001710.
%K A247497 nonn,tabl
%O A247497 0,4
%A A247497 _Peter Luschny_, Dec 13 2014