A178655 Triangle which contains the first differences of the Catalan triangle A001263 constructed along rows.

1, 1, -1, 1, 0, -1, 1, 2, -2, -1, 1, 5, 0, -5, -1, 1, 9, 10, -10, -9, -1, 1, 14, 35, 0, -35, -14, -1, 1, 20, 84, 70, -70, -84, -20, -1, 1, 27, 168, 294, 0, -294, -168, -27, -1, 1, 35, 300, 840, 588, -588, -840, -300, -35, -1

Offset

0,8

Links

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

Formula

T(n,k) = -T(n,n-k), n > 0.

T(n,k) = A001263(n,k+1) - A001263(n,k), n > 0. - R. J. Mathar, Jun 16 2015

Example

Triangle begins
  1;
  1,   -1;
  1,    0,   -1;
  1,    2,   -2,   -1;
  1,    5,    0,   -5,   -1;
  1,    9,   10,  -10,   -9,   -1;
  1,   14,   35,    0,  -35,  -14,   -1;
  1,   20,   84,   70,  -70,  -84,  -20,   -1;
  1,   27,  168,  294,    0, -294, -168,  -27,   -1;
  1,   35,  300,  840,  588, -588, -840, -300,  -35,   -1;

Mathematica

Join[{1}, Table[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial[n, k]^2, {n, 1, 10}, {k, 0, n}]//Flatten] (* G. C. Greubel, Jan 28 2019 *)

Prog

(PARI) {T(n, k) = if(n==0, 1, ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2)};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jan 28 2019
(Magma) [[n le 0 select 1 else ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 28 2019
(Sage) [1] + [[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2 for k in (0..n)] for n in (1..10)] # G. C. Greubel, Jan 28 2019

Crossrefs

Cf. A001263, A000007 (row sums).

Sequence in context: A361014 A064552 A209543 * A337278 A178304 A123585

Adjacent sequences: A178652 A178653 A178654 * A178656 A178657 A178658

Keyword

sign,tabl,easy

Author

Roger L. Bagula, Jun 01 2010

Status

approved