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

0,4

Links

Table of n, a(n) for n=0..44.

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.

Example

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,...

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);

Crossrefs

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

Sequence in context: A082771 A127157 A236406 * A202714 A022662 A295703

Adjacent sequences: A247494 A247495 A247496 * A247498 A247499 A247500

Keyword

nonn,tabl

Author

Peter Luschny, Dec 13 2014

Status

approved