A109954 Riordan array (1/(1+x)^3,x/(1+x)^2).

1, -3, 1, 6, -5, 1, -10, 15, -7, 1, 15, -35, 28, -9, 1, -21, 70, -84, 45, -11, 1, 28, -126, 210, -165, 66, -13, 1, -36, 210, -462, 495, -286, 91, -15, 1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1, 66, -715, 3003, -6435, 8008, -6188, 3060, -969, 190, -21, 1

Offset: 0

Author

Paul Barry, Jul 06 2005

Keywords

• easy
• sign
• tabl

Comments

Inverse of Riordan array (c(x)^3,x*c(x)^2) or A050155, with c(x) the g.f. of A000108. Unsigned array is the Riordan array (1/(1-x)^3,x/(1-x)^2), with T(n,k) = binomial(n+k+2,2*k+2).

Triangle of coefficients of polynomials defined by: c0=1; p(x, n) = (2 + c0 - x)*p(x, n - 1) + (-1 - c0 (2 - x))*p(x, n - 2) + c0*p(x, n - 3). Setting c0=0 gives A136674. - Roger L. Bagula, Apr 08 2008

The triangle entries Ts(n,k):=(-1)^(n-1)*A109954(n-1, k) = ((-1)^k)*binomial(n+k+1, 2(k+1)), n>=1, k=0..n-1, are the coefficients of x^(2*k) of the polynomial P(n,x^2) := (1 - (-1)^n*S(2*n,x))/x^2, with the Chebyshev S-polynomials with coefficient triangle given in A049310.

P(n,x^2) = - R(n+1,x)*S(n-1,x)/x^2 if n is even and P(n,x^2) = R(n,x)*S(n,x)/x^2 if n is odd, with R the monic integer Chebyshev T-polynomials with coefficient triangle given in A127672. - Wolfdieter Lang, Oct 24 2012.

Examples

			Triangle T(n, k) begins:
  n/k   0     1      2     3     4      5     6    7   8   9 10
  0:    1
  1:   -3     1
  2:    6    -5      1
  3:  -10    15     -7     1
  4:   15   -35     28    -9     1
  5:  -21    70    -84    45   -11      1
  6:   28  -126    210  -165    66    -13     1
  7:  -36   210   -462   495  -286     91   -15    1
  8:   45  -330    924 -1287  1001   -455   120  -17   1
  9:  -55   495  -1716  3003 -3003   1820  -680  153 -19   1
  10:  66  -715   3003 -6435  8008  -6188  3060 -969 190 -21  1
  ... Reformatted and extended by Wolfdieter Lang, Oct 24 2012.
		

Crossrefs

Cf. A129818, A085478.

Programs

Mathematica

c0 = 1; p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 2 - x + c0; p[x_, n_] := p[x, n] = (2 + c0 - x)*p[x, n - 1] + (-1 - c0 (2 - x))*p[x, n - 2] + c0*p[x, n - 3]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] - Roger L. Bagula, Apr 08 2008

Formula

Number triangle T(n, k) = (-1)^(n+k)*binomial(n+k+2, 2*k+2) [offset (0, 0)].