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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.

%I A176861 #10 Feb 08 2021 05:25:34
%S A176861 1,-6,-6,36,64,36,-240,-600,-600,-240,1800,5760,8100,5760,1800,-15120,
%T A176861 -58800,-105840,-105840,-58800,-15120,141120,645120,1411200,1806336,
%U A176861 1411200,645120,141120,-1451520,-7620480,-19595520,-30481920,-30481920,-19595520,-7620480,-1451520
%N 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.
%C A176861 Row sums are: 1, -12, 136, -1680, 23220, -359520, 6201216, -118298880, ...
%D A176861 F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 270.
%H A176861 G. C. Greubel, <a href="/A176861/b176861.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A176861 T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2).
%F A176861 T(n, k) = (-1)^n * A132159(n+2, k+2) * A132159(n+2, n-k+2). - _G. C. Greubel_, Feb 07 2021
%e A176861 Triangle begins as:
%e A176861        1;
%e A176861       -6,     -6;
%e A176861       36,     64,      36;
%e A176861     -240,   -600,    -600,    -240;
%e A176861     1800,   5760,    8100,    5760,    1800;
%e A176861   -15120, -58800, -105840, -105840,  -58800, -15120;
%e A176861   141120, 645120, 1411200, 1806336, 1411200, 645120, 141120;
%t A176861 T[n_, k_]:= (-1)^n*(k+1)!*(n-k+1)!*Binomial[n+2, k+2]*Binomial[n+2, n-k+2];
%t A176861 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
%o A176861 (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
%o A176861 (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
%Y A176861 Cf. A132159.
%K A176861 sign,tabl,easy,less
%O A176861 0,2
%A A176861 _Roger L. Bagula_, Apr 27 2010
%E A176861 Edited by _G. C. Greubel_, Feb 07 2021