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