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A176861
Triangle T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2) read by rows.
1
1, -6, -6, 36, 64, 36, -240, -600, -600, -240, 1800, 5760, 8100, 5760, 1800, -15120, -58800, -105840, -105840, -58800, -15120, 141120, 645120, 1411200, 1806336, 1411200, 645120, 141120, -1451520, -7620480, -19595520, -30481920, -30481920, -19595520, -7620480, -1451520
OFFSET
0,2
COMMENTS
Row sums are: 1, -12, 136, -1680, 23220, -359520, 6201216, -118298880, ...
REFERENCES
F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 270.
FORMULA
T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2).
T(n, k) = (-1)^n * A132159(n+2, k+2) * A132159(n+2, n-k+2). - G. C. Greubel, Feb 07 2021
EXAMPLE
Triangle begins as:
1;
-6, -6;
36, 64, 36;
-240, -600, -600, -240;
1800, 5760, 8100, 5760, 1800;
-15120, -58800, -105840, -105840, -58800, -15120;
141120, 645120, 1411200, 1806336, 1411200, 645120, 141120;
MATHEMATICA
T[n_, k_]:= (-1)^n*(k+1)!*(n-k+1)!*Binomial[n+2, k+2]*Binomial[n+2, n-k+2];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Sage) flatten([[(-1)^n*factorial(k+1)*factorial(n-k+1)*binomial(n+2, k+2)*binomial(n+2, n-k+2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021
(Magma) [(-1)^n*Factorial(k+1)*Factorial(n-k+1)*Binomial(n+2, k+2)*Binomial(n+2, n-k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021
CROSSREFS
Cf. A132159.
Sequence in context: A056454 A056452 A183622 * A377204 A245131 A038260
KEYWORD
sign,tabl,easy,less
AUTHOR
Roger L. Bagula, Apr 27 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 07 2021
STATUS
approved