A158193 Irregular triangle T(n, k) = ((-1)^(k+1)/2)*Sum_{j=0..n-2*k} binomial(n+2, j)*binomial(n+2, j+k+1)* binomial(n+2, j+2*k+2), read by rows.

-1, -9, -72, 3, -550, 50, -4140, 585, -10, -31017, 5880, -245, -232288, 54488, -3808, 35, -1742148, 480816, -47880, 1134, -13095450, 4110750, -532350, 22050, -126, -98687600, 34397880, -5466780, 333960, -5082, -745652160, 283510260, -53143200, 4348377, -118800, 462

Offset

0,2

Links

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

Matjaz Konvalinka, An inverse matrix formula in the right-quantum algebra, Electron. J. Combin. 15 (2008), R23.

Formula

T(n, k) = ((-1)^(k+1)/2)*Sum_{j=0..n-2*k} binomial(n+2, j)*binomial(n+2, j+k+1)* binomial(n+2, j+2*k+2).

Example

Irregular triangle begins as:
          -1;
          -9;
         -72,         3;
        -550,        50;
       -4140,       585,       -10;
      -31017,      5880,      -245;
     -232288,     54488,     -3808,      35;
    -1742148,    480816,    -47880,    1134;
   -13095450,   4110750,   -532350,   22050,    -126;
   -98687600,  34397880,  -5466780,  333960,   -5082;
  -745652160, 283510260, -53143200, 4348377, -118800, 462;

Mathematica

Table[Sum[(-1)^(k+1)*Binomial[n+2, j]*Binomial[n+2, j+k+1]*Binomial[n+2, j+2*k+2], {j, 0, n-2*k}]/2, {n, 0, 10}, {k, 0, Floor[n/2]}]//Flatten

Prog

(Magma)
A158193:= func< n, k | ((-1)^(k+1)/2)*(&+[Binomial(n+2, j)*Binomial(n+2, j+k+1)*Binomial(n+2, j+2*k+2): j in [0..n-2*k]]) >;
[A158193(n, k): k in [0..Floor(n/2)], n in [0..12]]; // G. C. Greubel, Jun 26 2021
(Sage)
def A158193(n, k): return ((-1)^(k+1)/2)*sum( binomial(n+2, j)*binomial(n+2, j+k+1)*binomial(n+2, j+2*k+2) for j in (0..n-2*k) )
flatten([[A158193(n, k) for k in (0..n//2)] for n in (0..12)]) # G. C. Greubel, Jun 26 2021

Crossrefs

Sequence in context: A356239 A324413 A144745 * A123987 A003365 A044196

Adjacent sequences: A158190 A158191 A158192 * A158194 A158195 A158196

Keyword

sign,tabf

Author

Roger L. Bagula, Mar 13 2009

Extensions

Edited by G. C. Greubel, Jun 26 2021

Status

approved