A159856 Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.

1, 2, -1, 3, -4, 1, 4, -11, 6, -1, 5, -26, 23, -8, 1, 6, -57, 72, -39, 10, -1, 7, -120, 201, -150, 59, -12, 1, 8, -247, 522, -501, 268, -83, 14, -1, 9, -502, 1291, -1524, 1037, -434, 111, -16, 1, 10, -1013, 3084, -4339, 3598, -1905, 656, -143, 18, -1

Offset

0,2

Comments

A Riordan array - see the Luzon references.

The second column is A000295 signed. - Michel Marcus, Feb 14 2014

Links

Table of n, a(n) for n=0..54.

Ana Luzón, Iterative Processes Related to Riordan Arrays: The Reciprocation and the Inversion of Power Series, arXiv:0907.2328 [math.CO]; Discrete Math., 310 (2010), 3607-3618.

Ana Luzón and Manuel A. Morón, Riordan matrices in the reciprocation of quadratic polynomials, Linear Algebra Appl. 430 (2009), no. 8-9, 22542270.

Formula

From R. J. Mathar, May 31 2009: (Start)

Sum_{k=0..n} T(n,k) = A080956(n).

Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)

T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k-1) for k > 0. - Vladimir Kruchinin, Nov 22 2016 [corrected by Werner Schulte, May 09 2024]

G.f.: (1-2*x)/(1-x)^2/(1-2*x+y*x). - Vladimir Kruchinin, Nov 22 2016

Example

Triangle begins
  1;
  2,   -1;
  3,   -4,    1;
  4,  -11,    6,   -1;
  5,  -26,   23,   -8,    1;
  6,  -57,   72,  -39,   10,   -1;
  7, -120,  201, -150,   59,  -12,    1;
  ...

Mathematica

With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
+y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)

Prog

(Maxima)
T(n, k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k, x, 0, 15), x, n); /* Vladimir Kruchinin, Nov 22 2016 */

Crossrefs

Cf. A000295, A047926, A080956, A181690.

Sequence in context: A247239 A198060 A327084 * A137649 A180915 A240783

Adjacent sequences: A159853 A159854 A159855 * A159857 A159858 A159859

Keyword

easy,sign,tabl

Author

Philippe Deléham, Apr 24 2009

Extensions

a(41) corrected by Georg Fischer, Feb 18 2020

Status

approved