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Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
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%I #20 Nov 10 2013 03:51:11

%S 1,1,0,1,1,0,1,3,0,0,1,6,1,0,0,1,10,5,0,0,0,1,15,15,1,0,0,0,1,21,35,7,

%T 0,0,0,0,1,28,70,28,1,0,0,0,0,1,36,126,84,9,0,0,0,0,0,1,45,210,210,45,

%U 1,0,0,0,0,0

%N Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%C Riordan array (1/(1-x), x^2/(1-x)^2).

%C A skewed version of triangular array A085478.

%C Mirror image of triangle in A098158.

%C Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.

%C Sum_{k, 0<=k<=n} T(n,k)*x^(n-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 = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

%F T(n,k) = binomial(n,2k).

%F G.f.: (1-x)/((1-x)^2-y*x^2).

%F T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if n<k.

%F T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - _Philippe Deléham_, Nov 10 2013

%e Triangle begins :

%e 1

%e 1, 0

%e 1, 1, 0

%e 1, 3, 0, 0

%e 1, 6, 1, 0, 0

%e 1, 10, 5, 0, 0, 0

%e 1, 15, 15, 1, 0, 0, 0

%e 1, 21, 35, 7, 0, 0, 0, 0

%e 1, 28, 70, 28, 1, 0, 0, 0, 0

%Y Cf. A007318, A011782 (row sums), A034839, A098158.

%K nonn,tabl

%O 0,8

%A _Philippe Deléham_, Dec 10 2011