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%I #34 Jan 22 2020 02:42:06
%S 1,2,1,5,6,2,13,24,15,3,34,84,78,32,5,89,275,340,210,65,8,233,864,
%T 1335,1100,510,126,13,610,2639,4893,5040,3115,1155,238,21,1597,7896,
%U 17080,21112,16310,8064,2492,440,34,4181,23256,57492,82908,76860,47502,19572,5184,801,55
%N A binomial transform of Fibonacci numbers.
%C Triangle begins:
%C 1,
%C 2, 1,
%C 5, 6, 2,
%C 13, 24, 15, 3,
%C 34, 84, 78, 32, 5,
%C 89, 275, 340, 210, 65, 8,
%C 233, 864, 1335, 1100, 510, 126, 13,
%C 610, 2639, 4893, 5040, 3115, 1155, 238, 21,
%C 1597, 7896, 17080, 21112, 16310, 8064, 2492, 440, 34,
%C ...
%C Diagonal: a(n,n) = F(n+1).
%C First column: a(n,0) = F(2n+1) (A001519).
%C Row sums: Sum_{k=0..n} a(n,k) = F(3n+1) (A033887).
%C Alternated row sums: Sum_{k=0..n} (-1)^k * a(n,k) = 1.
%C Diagonal sums: Sum_{k=0..floor(n/2)} a(n-k,k) = A208481(n).
%C Alternated diagonal sums: Sum_{k=0..floor(n/2)} (-1)^k * a(n-k,k) = F(n+3)-1 (A000071).
%C Row square-sums: Sum_{k=0..n} a(n,k)^2 = A208588(n).
%C Central coefficients: a(2*n,n) = binomial(2n,n)*F(3n+1) (A208473), where F(n) are the Fibonacci numbers (A000045).
%C Mirror image of the triangle in A122070. - _Philippe Deléham_, Mar 13 2012
%C Subtriangle of (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 13 2012
%H G. C. Greubel, <a href="/A185384/b185384.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F a(n,k) = Sum_{i=k..n} binomial(n,i)*binomial(i,k)*F(i+1).
%F a(n,k) = binomial(n,i) * Sum_{i=k..n} binomial(n-k,n-i)*F(i+1).
%F Explicit form: a(n,k) = binomial(n,k)*F(2*n-k+1).
%F G.f.: (1-x)/(1-3*x+x^2-x*y-x^2*y-x^2*y^2).
%F Recurrence: a(n+2,k+2) = 3*a(n+1,k+2) + a(n+1,k+1) - a(n,k+2) + a(n,k+1) + a(n,k).
%F T(n,k) = A122070(n,n-k). - _Philippe Deléham_, Mar 13 2012
%e From _Philippe Deléham_, Mar 13 2012: (Start)
%e (1, 1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 2, 1, 0;
%e 5, 6, 2, 0;
%e 13, 24, 15, 3, 0;
%e 34, 84, 78, 32, 5, 0;
%e 89, 275, 340, 210, 65, 8, 0;
%e 233, 864, 1335, 1100, 510, 126, 13, 0;
%e ... (End)
%t Flatten[Table[Sum[Binomial[n,i]Binomial[i,k]Fibonacci[i+1],{i,k,n}],{n,0,20},{k,0,n}]]
%t CoefficientList[Series[CoefficientList[Series[(1 - x)/(1 - 3*x + x^2 - x*y - x^2*y - x^2*y^2), {x, 0, 10}], x], {y, 0, 10}], y] // Flatten (* _G. C. Greubel_, Jun 28 2017 *)
%o (Maxima) create_list(binomial(n,k)*fib(2*n-k+1),n,0,20,k,0,n);
%o (PARI) for(n=0,10, for(k=0,n, print1(sum(i=k,n, binomial(n,i) * binomial(i,k) * fibonacci(i+1)), ", "))) \\ _G. C. Greubel_, Jun 28 2017
%Y Cf. A000045, A001519, A033887, A208481, A000071, A208588, A208473.
%Y Cf. A122070.
%K nonn,tabl,easy
%O 0,2
%A _Emanuele Munarini_, Feb 29 2012
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