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%I #18 Sep 08 2022 08:45:28
%S 1,1,2,2,6,5,3,15,24,13,5,32,78,84,34,8,65,210,340,275,89,13,126,510,
%T 1100,1335,864,233,21,238,1155,3115,5040,4893,2639,610,34,440,2492,
%U 8064,16310,21112,17080,7896,1597,55,801,5184,19572,47502,76860,82908,57492,23256,4181
%N Triangle given by T(n,k) = Fibonacci(n+k+1)*binomial(n,k) for 0<=k<=n.
%C Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C Mirror image of the triangle in A185384.
%H G. C. Greubel, <a href="/A122070/b122070.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = A000045(n+k+1)*A007318(n,k) .
%F T(n,n) = Fibonacci(2*n+1) = A001519(n+1) .
%F Sum_{k=0..n} T(n,k) = Fibonacci(3*n+1) = A033887(n) .
%F Sum_{k=0..n}(-1)^k*T(n,k) = (-1)^n = A033999(n) .
%F Sum_{k=0..floor(n/2)} T(n-k,k) = (Fibonacci(n+1))^2 = A007598(n+1).
%F Sum_{k=0..n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n).
%F Sum_{k=0..n} T(n,k)^2 = A208588(n).
%F G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2).
%F T(n,k) = T(n-1,k) + 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
%F T(n,k) = A185384(n,n-k).
%F T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n).
%e Triangle begins:
%e 1;
%e 1, 2;
%e 2, 6, 5;
%e 3, 15, 24, 13;
%e 5, 32, 78, 84, 34;
%e 8, 65, 210, 340, 275, 89;
%e 13, 126, 510, 1100, 1335, 864, 233;
%e (0, 1, 1, -1, 0, 0, ...) DELTA (1, 1, 1, 0, 0, ...) begins :
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 2, 6, 5;
%e 0, 3, 15, 24, 13;
%e 0, 5, 32, 78, 84, 34;
%e 0, 8, 65, 210, 340, 275, 89;
%e 0, 13, 126, 510, 1100, 1335, 864, 233;
%p with(combinat): seq(seq(binomial(n,k)*fibonacci(n+k+1), k=0..n), n=0..10); # _G. C. Greubel_, Oct 02 2019
%t Table[Fibonacci[n+k+1]*Binomial[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 02 2019 *)
%o (PARI) T(n,k) = binomial(n,k)*fibonacci(n+k+1);
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 02 2019
%o (Magma) [Binomial(n,k)*Fibonacci(n+k+1): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Oct 02 2019
%o (Sage) [[binomial(n,k)*fibonacci(n+k+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Oct 02 2019
%o (GAP) Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n+ k+1) ))); # _G. C. Greubel_, Oct 02 2019
%Y Cf. A000045, A001519, A033887, A033889, A185384.
%K nonn,tabl
%O 0,3
%A _Philippe Deléham_, Oct 15 2006, Mar 13 2012
%E Corrected and extended by _Philippe Deléham_, Mar 13 2012
%E Term a(50) corrected by _G. C. Greubel_, Oct 02 2019
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