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A182533 A symmetrical triangle. Read by rows: T(n,k) = 2*C(n-2,k-1) - C(n-2,k) - C(n-2,k-2), n >= 2, 0 <= k <= n, with T(0,0) = 0, T(1,0) = T(1,1) = 1.

%I A182533 #58 Nov 29 2018 10:40:37
%S A182533 0,1,1,-1,2,-1,-1,1,1,-1,-1,0,2,0,-1,-1,-1,2,2,-1,-1,-1,-2,1,4,1,-2,
%T A182533 -1,-1,-3,-1,5,5,-1,-3,-1,-1,-4,-4,4,10,4,-4,-4,-1,-1,-5,-8,0,14,14,0,
%U A182533 -8,-5,-1,-1,-6,-13,-8,14,28,14,-8,-13,-6,-1,-1,-7,-19,-21,6,42,42,6,-21,-19,-7,-1
%N A182533 A symmetrical triangle. Read by rows:  T(n,k) = 2*C(n-2,k-1) - C(n-2,k) - C(n-2,k-2), n >= 2, 0 <= k <= n, with T(0,0) = 0, T(1,0) = T(1,1) = 1.
%H A182533 Muniru A Asiru, <a href="/A182533/b182533.txt">Table of n, a(n) for n = 1..5151</a> (rows n=1..100)
%H A182533 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H A182533 <a href="http://oeis.org/w/images/7/74/Triengel3.jpg">Graphical representation</a>
%F A182533 T(n,k) = 2*C(n-2,k-1) - C(n-2,k) - C(n-2,k-2).
%F A182533 Row sums = 0, for n >= 2.
%F A182533 T(2*n,n) = 2*A000108(n-1).
%F A182533 T(2*n+3,n) = -A120009(n).
%F A182533 T(n+10,3) = -A137742(n+2).
%F A182533 Sum_{k=0..n} T(n,k)*2^k = -(3^(n-2)) where n >= 2.
%F A182533 Sum_{k=0..n} T(n,k)*3^k = -(4^(n-1)) where n >= 2.
%F A182533 Sum_{k=0..n} T(n,k)*p^k = -((p+1)^(n-2))*((p-1)^2) where n >= 2.
%e A182533 Triangle begins
%e A182533    0;
%e A182533    1,  1;
%e A182533   -1,  2, -1;
%e A182533   -1,  1,  1, -1;
%e A182533   -1,  0,  2,  0, -1;
%e A182533   -1, -1,  2,  2, -1, -1;
%e A182533   -1, -2,  1,  4,  1, -2, -1;
%e A182533   -1, -3, -1,  5,  5, -1, -3, -1;
%e A182533   -1, -4, -4,  4, 10,  4, -4, -4, -1;
%t A182533 Flatten[Join[{{0}, {1, 1}}, Table[2*Binomial[n - 2, k - 1] - Binomial[n - 2, k] - Binomial[n - 2, k - 2], {n, 2, 15}, {k, 0, n}]]] (* _T. D. Noe_, May 08 2012 *)
%o A182533 (GAP) T:=Concatenation([0,1,1],Flat(List([2..11],n->List([0..n],k->2*Binomial(n-2,k-1)-Binomial(n-2,k)-Binomial(n-2,k-2))))); # _Muniru A Asiru_, Nov 29 2018
%Y A182533 Cf. A000108, A120009, A137742.
%K A182533 sign,tabl
%O A182533 1,5
%A A182533 _Alzhekeyev Ascar M_, May 04 2012