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%I #22 Jan 18 2017 22:19:50
%S 1,1,1,2,3,1,4,9,6,1,8,24,25,10,1,16,60,85,55,15,1,32,144,258,231,105,
%T 21,1,64,336,728,833,532,182,28,1,128,768,1952,2720,2241,1092,294,36,
%U 1,256,1728,5040,8280,8361,5301,2058,450,45,1
%N Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C Rows sums: A006012; Diagonal sums: A052960.
%C The sums of each column of A117317 with its subsequent column, treated as a lower triangular matrix with an initial null column attached, or, equivalently, the products of the row polynomials p(n,y) of A117317 with (1+y) with the initial first row below added to the final result. The reversal of A117317 is A056242 with several combinatorial interpretations. - _Tom Copeland_, Jan 08 2017
%F Sum_{k=0..n} T(n,k)*x^k = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A123335(n), A000007(n), A000012(n), A006012(n), A084120(n), A090965(n), A165225(n), A165229(n), A165230(n), A165231(n), A165232(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively.
%F G.f.: (1-(1+y)*x)/(1-2(1+y)*x+(y+y^2)*x^2). - _Philippe Deléham_, Dec 19 2011
%F T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n<k. - _Philippe Deléham_, Dec 19 2011
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 4, 9, 6, 1;
%e 8, 24, 25, 10, 1; ...
%Y Cf. A000012, A000217, A005582, A011782, A084858.
%Y Cf. A056242, A117317.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Sep 09 2009
%E O.g.f. corrected by _Tom Copeland_, Jan 15 2017
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