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A171631 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.

%I A171631 #13 Oct 02 2018 11:15:26
%S A171631 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 A171631 560,315,84,7,64,392,1008,1400,1120,504,112,8,81,576,1764,3024,3150,
%U A171631 2016,756,144,9,100,810,2880,5880,7560,6300,3360,1080,180,10,121,1100
%N A171631 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 A171631 If T(0,0) = 0 is prepended, then row sums give A001788.
%D A171631 Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.
%F A171631 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 A171631 From _Franck Maminirina Ramaharo_, Oct 02 2018: (Start)
%F A171631 T(n,1) = A000290(n).
%F A171631 T(n,2) = A011379(n).
%F A171631 T(n,3) = 3*A002417(n-2).
%F A171631 T(n,n-2) = A046092(n-1).
%F A171631 T(n,n-3) = 9*A000292(n-2).
%F A171631 G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)
%e A171631 Triangle begins:
%e A171631 n\k|  0    1     2     3     4     6    7    8  9
%e A171631 -------------------------------------------------
%e A171631 1  |  1
%e A171631 2  |  4    2
%e A171631 3  |  9   12     3
%e A171631 4  | 16   36    24     4
%e A171631 5  | 25   80    90    40     5
%e A171631 6  | 36  150   240   180    60     6
%e A171631 7  | 49  252   525   560   315    84    7
%e A171631 8  | 64  392  1008  1400  1120   504  112    8
%e A171631 9  | 81  576  1764  3024  3150  2016  756  144  9
%e A171631 ... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018
%t A171631 Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten
%o A171631 (Maxima) T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
%o A171631 tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* _Franck Maminirina Ramaharo_, Oct 02 2018 */
%Y A171631 Cf. A003506, A007318, A127952, A171531.
%K A171631 nonn,tabl,easy
%O A171631 1,2
%A A171631 _Roger L. Bagula_, Dec 13 2009
%E A171631 Edited and new name by _Franck Maminirina Ramaharo_, Oct 02 2018