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A binomial transform of Fibonacci numbers.
7

%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