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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). 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 24 15:25 EDT 2019. Contains 323532 sequences. (Running on oeis4.)