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%I #13 Oct 02 2018 11:15:26
%S 1,4,2,9,12,3,16,36,24,4,25,80,90,40,5,36,150,240,180,60,6,49,252,525,
%T 560,315,84,7,64,392,1008,1400,1120,504,112,8,81,576,1764,3024,3150,
%U 2016,756,144,9,100,810,2880,5880,7560,6300,3360,1080,180,10,121,1100
%N Triangle read by rows: T(n,k) = n*(binomial(n-2, k-1) + n*binomial(n-2, k)), n > 0 and 0 <= k <= n - 1.
%C If T(0,0) = 0 is prepended, then row sums give A001788.
%D Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.
%F Let p(x;n) = (x + 1)^n. Then row n are the coefficients in the expansion of p''(x;n) - x*p'(x;n) + n*p(x;n) = n*(x + n)*(x + 1)^(n - 2).
%F From _Franck Maminirina Ramaharo_, Oct 02 2018: (Start)
%F T(n,1) = A000290(n).
%F T(n,2) = A011379(n).
%F T(n,3) = 3*A002417(n-2).
%F T(n,n-2) = A046092(n-1).
%F T(n,n-3) = 9*A000292(n-2).
%F G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)
%e Triangle begins:
%e n\k| 0 1 2 3 4 6 7 8 9
%e -------------------------------------------------
%e 1 | 1
%e 2 | 4 2
%e 3 | 9 12 3
%e 4 | 16 36 24 4
%e 5 | 25 80 90 40 5
%e 6 | 36 150 240 180 60 6
%e 7 | 49 252 525 560 315 84 7
%e 8 | 64 392 1008 1400 1120 504 112 8
%e 9 | 81 576 1764 3024 3150 2016 756 144 9
%e ... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018
%t Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten
%o (Maxima) T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
%o tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* _Franck Maminirina Ramaharo_, Oct 02 2018 */
%Y Cf. A003506, A007318, A127952, A171531.
%K nonn,tabl,easy
%O 1,2
%A _Roger L. Bagula_, Dec 13 2009
%E Edited and new name by _Franck Maminirina Ramaharo_, Oct 02 2018