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Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
3

%I #26 Aug 21 2016 14:19:30

%S 1,0,3,0,1,5,0,0,5,7,0,0,1,14,9,0,0,0,7,30,11,0,0,0,1,27,55,13,0,0,0,

%T 0,9,77,91,15,0,0,0,0,1,44,182,140,17,0,0,0,0,0,11,156,378,204,19,0,0,

%U 0,0,0,1,65,450,714,285,21,0

%N Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

%C Previous name was: "A scaled version of triangle A082985."

%C Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%H R. Zielinski, <a href="https://arxiv.org/abs/1608.04006">Induction and Analogy in a Problem of Finite Sums</a> arXiv:1608.04006 [math.GM], 2016.

%F T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.

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

%F Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).

%F Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.

%F Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.

%F From _Peter Bala_, Aug 17 2016: (Start)

%F Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:

%F Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.

%F For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.

%F There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

%e Triangle begins :

%e 1

%e 0, 3

%e 0, 1, 5

%e 0, 0, 5, 7

%e 0, 0, 1, 14, 9

%e 0, 0, 0, 7, 30, 11

%e 0, 0, 0, 1, 27, 55, 13

%e 0, 0, 0, 0, 9, 77, 91, 15

%e 0, 0, 0, 0, 1, 44, 182, 140, 17

%e 0, 0, 0, 0, 0, 11, 156, 378, 204, 19

%e 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21

%e 0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

%p s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):

%p seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # _Peter Luschny_, Aug 17 2016

%Y Cf. A082985 which is another version of this triangle.

%Y Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

%K easy,nonn,tabl

%O 0,3

%A _Philippe Deléham_, Feb 24 2012

%E New name using a formula of the author from _Peter Luschny_, Aug 17 2016