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A016095 Triangular array T(n,k) read by rows, where T(n,k) = coefficient of x^n*y^k in 1/(1-x-y-(x+y)^2).

%I A016095 #38 Dec 30 2024 21:52:29
%S A016095 1,1,1,2,4,2,3,9,9,3,5,20,30,20,5,8,40,80,80,40,8,13,78,195,260,195,
%T A016095 78,13,21,147,441,735,735,441,147,21,34,272,952,1904,2380,1904,952,
%U A016095 272,34,55,495,1980,4620,6930,6930,4620,1980,495,55
%N A016095 Triangular array T(n,k) read by rows, where T(n,k) = coefficient of x^n*y^k in 1/(1-x-y-(x+y)^2).
%C A016095 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 10 2005
%F A016095 G.f.: 1/(1-x-y-(x+y)^2).
%F A016095 T(n,k) = Fibonacci(n+1)*binomial(n,k) = A000045(n+1)*A007318(n,k). - _Philippe Deléham_, Oct 14 2006
%F A016095 Sum_{k=0..floor(n/2)} T(n-k,k) = A123392(n). - _Philippe Deléham_, Oct 14 2006
%F A016095 G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x*(1+y)/((2*k+2+ x*(1+y))*x*(1+y)+ 1/T(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Nov 06 2013
%F A016095 T(n,k) = T(n-1,k)+T(n-1,k-1)+T(n-2,k)+2*T(n-2,k-1)+T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = T(2,2) = 2, T(2,1) = 4, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Nov 12 2013
%e A016095 Triangle begins:
%e A016095   1;
%e A016095   1,  1;
%e A016095   2,  4,  2;
%e A016095   3,  9,  9,  3;
%e A016095   5, 20, 30, 20,  5;
%e A016095   8, 40, 80, 80, 40, 8;
%e A016095   ...
%p A016095 read transforms; 1/(1-x-y-(x+y)^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);
%t A016095 T[n_, k_] := SeriesCoefficient[1/(1-x-y-(x+y)^2), {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 04 2017 *)
%Y A016095 Columns include A000045, A023607. Central diagonal is A102307. Antidiagonal sums are in A063727.
%K A016095 nonn,tabl,easy
%O A016095 0,4
%A A016095 _N. J. A. Sloane_, Jan 23 2001