A104558 Triangle, read by rows, equal to the matrix inverse of A104557 and related to Laguerre polynomials.

1, -1, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 6, -6, 1, 0, 0, -6, 18, -9, 1, 0, 0, 0, -24, 36, -12, 1, 0, 0, 0, 24, -96, 72, -16, 1, 0, 0, 0, 0, 120, -240, 120, -20, 1, 0, 0, 0, 0, -120, 600, -600, 200, -25, 1, 0, 0, 0, 0, 0, -720, 1800, -1200, 300, -30, 1, 0, 0, 0, 0, 0, 720, -4320, 5400, -2400, 450, -36, 1, 0, 0, 0, 0, 0, 0, 5040, -15120, 12600, -4200, 630, -42, 1

Offset

0,5

Comments

Even-indexed rows are found in A066667 (generalized Laguerre polynomials). Odd-indexed rows are found in A021009 (Laguerre polynomials L_n(x)). Row sums equal A056920 (offset 1). Absolute row sums equal A056953 (offset 1).

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

Formula

T(n, k) = (-1)^(n-k)*(n-k)!*C(1+[n/2], k+1-[(n+1)/2])*C([(n+1)/2], k-[n/2]).

Example

Rows begin:
  1;
  -1,1;
  0,-2,1;
  0,2,-4,1;
  0,0,6,-6,1;
  0,0,-6,18,-9,1;
  0,0,0,-24,36,-12,1;
  0,0,0,24,-96,72,-16,1;
  0,0,0,0,120,-240,120,-20,1;
  0,0,0,0,-120,600,-600,200,-25,1;
  ...
Unsigned columns read downwards equals rows of matrix inverse A104557 read backwards:
  1;
  1,1;
  2,2,1;
  6,6,4,1;
  24,24,18,6,1;
  120,120,96,36,9,1;
  ...

Mathematica

T[n_, k_] := (-1)^(n - k)*(n - k)!*Binomial[1 + Floor[n/2], k + 1 - Floor[(n + 1)/2]]*Binomial[Floor[(n+1)/2], k -Floor[n/2]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2018 *)

Prog

(PARI) {T(n, k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2, k+1-(n+1)\2)* binomial( (n+1)\2, k-n\2)};
(Magma) /* As triangle */ [[(-1)^(n-k)*Factorial(n-k)*Binomial(1+ Floor(n/2), k +1 -Floor((n+1)/2))*Binomial(Floor((n+1)/2), k - Floor(n/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 14 2018

Crossrefs

Cf. A104557, A066667, A021009, A056920, A056953.

Sequence in context: A110280 A061009 A144106 * A206022 A115247 A204163

Adjacent sequences: A104555 A104556 A104557 * A104559 A104560 A104561

Keyword

sign,tabl

Author

Paul D. Hanna, Mar 16 2005

Status

approved