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A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A183190 #28 Nov 12 2019 10:09:05
%S A183190 1,1,0,2,1,0,4,4,1,0,8,12,6,1,0,16,32,24,8,1,0,32,80,80,40,10,1,0,64,
%T A183190 192,240,160,60,12,1,0,128,448,672,560,280,84,14,1,0,256,1024,1792,
%U A183190 1792,1120,448,112,16,1,0,512,2304,4608,5376,4032,2016,672,144,18,1,0
%N A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A183190 A071919*A007318 as infinite lower triangular matrices.
%C A183190 A129186*A038207 as infinite lower triangular matrices.
%C A183190 From _Paul Curtz_, Nov 12 2019: (Start)
%C A183190 If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:
%C A183190    The first column is A166444.
%C A183190    The second column is A139756.
%C A183190    The antidiagonal sums are A000129 (Pell numbers).
%C A183190    The row sums are (-1)^n*A141413.
%C A183190    The signed row sums are 0 followed by 1's, autosequence companion to A054977.
%C A183190 (End)
%F A183190 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .
%F A183190 G.f.: (1-(1+y)*x)/(1-(2+y)*x).
%F A183190 Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.
%F A183190 Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.
%F A183190 T(2n,n) = A069720(n).
%e A183190 Triangle begins:
%e A183190    1;
%e A183190    1,  0;
%e A183190    2,  1,  0;
%e A183190    4,  4,  1,  0;
%e A183190    8, 12,  6,  1,  0;
%e A183190   16, 32, 24,  8,  1, 0;
%e A183190   32, 80, 80, 40, 10, 1, 0;
%e A183190   ...
%p A183190 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A183190       `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
%p A183190     end:
%p A183190 seq(seq(T(n,k), k=0..n), n=0..12);  # _Alois P. Heinz_, Nov 08 2019
%t A183190 T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
%t A183190 T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[_, _] = 0;
%t A183190 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 08 2019 *)
%Y A183190 Essentially the same as A038207, A062715, A065109.
%Y A183190 Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).
%Y A183190 Cf. A141413, A166444, A054977.
%K A183190 nonn,tabl
%O A183190 0,4
%A A183190 _Philippe Deléham_, Dec 14 2011