A176860 Triangle, read by rows, T(n, k) = (-1)^k * (n-k+1)^(n+2) * binomial(n+1, k).

1, 8, -2, 81, -48, 3, 1024, -972, 192, -4, 15625, -20480, 7290, -640, 5, 279936, -468750, 245760, -43740, 1920, -6, 5764801, -11757312, 8203125, -2293760, 229635, -5376, 7, 134217728, -322828856, 282175488, -109375000, 18350080, -1102248, 14336, -8

Offset

0,2

References

F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 267.

Links

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

Formula

T(n, k) = (-1)^k * (n-k+1)^(n+2) * binomial(n+1, k).

Sum_{k=0..n} T(n, k) = (n + 1)*(n + 2)!/2 = A001286(n+2). - G. C. Greubel, Feb 07 2021

Example

Triangle begins as:
          1;
          8,         -2;
         81,        -48,         3;
       1024,       -972,       192,         -4;
      15625,     -20480,      7290,       -640,        5;
     279936,    -468750,    245760,     -43740,     1920,       -6;
    5764801,  -11757312,   8203125,   -2293760,   229635,    -5376,     7;
  134217728, -322828856, 282175488, -109375000, 18350080, -1102248, 14336, -8;

Mathematica

T[n_, k_]:= (-1)^k*(n-k+1)^(n+2)*Binomial[n+1, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage) flatten([[ (-1)^k*(n-k+1)^(n+2)*binomial(n+1, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021
(Magma) [(-1)^k*(n-k+1)^(n+2)*Binomial(n+1, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021

Crossrefs

Cf. A001286.

Sequence in context: A032761 A262896 A188898 * A281068 A093082 A107671

Adjacent sequences: A176857 A176858 A176859 * A176861 A176862 A176863

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 27 2010

Extensions

Edited by G. C. Greubel, Feb 07 2021

Status

approved