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A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

%I A207543 #26 Aug 21 2016 14:19:30
%S A207543 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 A207543 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 A207543 0,0,0,1,65,450,714,285,21,0
%N A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
%C A207543 Previous name was: "A scaled version of triangle A082985."
%C A207543 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 A207543 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 A207543 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 A207543 G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
%F A207543 Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).
%F A207543 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 A207543 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 A207543 From _Peter Bala_, Aug 17 2016: (Start)
%F A207543 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 A207543 Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.
%F A207543 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 A207543 There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)
%e A207543 Triangle begins :
%e A207543 1
%e A207543 0, 3
%e A207543 0, 1, 5
%e A207543 0, 0, 5, 7
%e A207543 0, 0, 1, 14, 9
%e A207543 0, 0, 0, 7, 30, 11
%e A207543 0, 0, 0, 1, 27, 55, 13
%e A207543 0, 0, 0, 0, 9, 77, 91, 15
%e A207543 0, 0, 0, 0, 1, 44, 182, 140, 17
%e A207543 0, 0, 0, 0, 0, 11, 156, 378, 204, 19
%e A207543 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21
%e A207543 0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23
%p A207543 s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):
%p A207543 seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # _Peter Luschny_, Aug 17 2016
%Y A207543 Cf. A082985 which is another version of this triangle.
%Y A207543 Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.
%K A207543 easy,nonn,tabl
%O A207543 0,3
%A A207543 _Philippe Deléham_, Feb 24 2012
%E A207543 New name using a formula of the author from _Peter Luschny_, Aug 17 2016