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A182001 Riordan array ((2*x+1)/(1-x-x^2), x/(1-x-x^2)).

%I A182001 #19 Sep 08 2022 08:45:54
%S A182001 1,3,1,4,4,1,7,9,5,1,11,20,15,6,1,18,40,40,22,7,1,29,78,95,68,30,8,1,
%T A182001 47,147,213,185,105,39,9,1,76,272,455,466,320,152,49,10,1,123,495,940,
%U A182001 1106,891,511,210,60,11,1,199,890,1890,2512,2317,1554,770,280,72,12,1
%N A182001 Riordan array ((2*x+1)/(1-x-x^2), x/(1-x-x^2)).
%C A182001 Subtriangle of the triangle given by (0, 3, -5/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A182001 Antidiagonal sums are A001045(n+2).
%H A182001 G. C. Greubel, <a href="/A182001/b182001.txt">Rows n = 0..100 of triangle, flattened</a>
%F A182001 G.f.: (1+2*x)/(1-x-y*x-x^2).
%F A182001 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 3, T(n,k) = 0 if k<0 or if k>n.
%F A182001 Sum_{k=0..nn} T(n,k)*x^k = A000034(n), A000032(n+1), A048654(n), A108300(n), A048875(n) for x = -1, 0, 1, 2, 3 respectively.
%e A182001 Triangle begins :
%e A182001     1;
%e A182001     3,   1;
%e A182001     4,   4,    1;
%e A182001     7,   9,    5,    1;
%e A182001    11,  20,   15,    6,    1;
%e A182001    18,  40,   40,   22,    7,    1;
%e A182001    29,  78,   95,   68,   30,    8,   1;
%e A182001    47, 147,  213,  185,  105,   39,   9,   1;
%e A182001    76, 272,  455,  466,  320,  152,  49,  10, 1;
%e A182001   123, 495,  940, 1106,  891,  511, 210,  60, 11,  1;
%e A182001   199, 890, 1890, 2512, 2317, 1554, 770, 280, 72, 12, 1;
%e A182001 (0, 3, -5/3, -1/3, 0, 0, ...) DELTA (1, 0, -2/3, 2/3, 0, 0, ...) begins:
%e A182001   1;
%e A182001   0,  1;
%e A182001   0,  3,  1;
%e A182001   0,  4,  4,  1;
%e A182001   0,  7,  9,  5,  1;
%e A182001   0, 11, 20, 15,  6, 1;
%e A182001   0, 18, 40, 40, 22, 7, 1;
%p A182001 with(combinat);
%p A182001 T:= proc(n, k) option remember;
%p A182001       if k<0 or k>n then 0
%p A182001     elif k=n then 1
%p A182001     elif k=0 then fibonacci(n+2) + fibonacci(n)
%p A182001     else T(n-1,k) + T(n-1,k-1) + T(n-2,k)
%p A182001       fi; end:
%p A182001 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Feb 18 2020
%t A182001 With[{m = 10}, CoefficientList[CoefficientList[Series[(1+2*x)/(1-x-y*x-x^2), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 18 2020 *)
%t A182001 T[n_, k_]:= T[n, k]= If[k<0||k>n, 0, If[k==n, 1, If[k==0, LucasL[n+1], T[n-1, k] + T[n-1, k-1] + T[n-2, k] ]]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 18 2020 *)
%o A182001 (Magma)
%o A182001 function T(n,k)
%o A182001   if k lt 0 or k gt n then return 0;
%o A182001   elif k eq n then return 1;
%o A182001   elif k eq 0 then return Lucas(n+1);
%o A182001   else return T(n-1,k) + T(n-1,k-1) + T(n-2,k);
%o A182001   end if; return T; end function;
%o A182001 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 18 2020
%Y A182001 Cf. Columns : A000032, A023607, A152881
%K A182001 easy,nonn,tabl
%O A182001 0,2
%A A182001 _Philippe Deléham_, Apr 05 2012
%E A182001 a(29) corrected by and a(55)-a(65) from _Georg Fischer_, Feb 18 2020