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URL: https://oeis.org/A185384

⇱ A185384 - OEIS

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A185384
A binomial transform of Fibonacci numbers.
7
1, 2, 1, 5, 6, 2, 13, 24, 15, 3, 34, 84, 78, 32, 5, 89, 275, 340, 210, 65, 8, 233, 864, 1335, 1100, 510, 126, 13, 610, 2639, 4893, 5040, 3115, 1155, 238, 21, 1597, 7896, 17080, 21112, 16310, 8064, 2492, 440, 34, 4181, 23256, 57492, 82908, 76860, 47502, 19572, 5184, 801, 55
OFFSET
0,2
COMMENTS
Triangle begins:
1,
2, 1,
5, 6, 2,
13, 24, 15, 3,
34, 84, 78, 32, 5,
89, 275, 340, 210, 65, 8,
233, 864, 1335, 1100, 510, 126, 13,
610, 2639, 4893, 5040, 3115, 1155, 238, 21,
1597, 7896, 17080, 21112, 16310, 8064, 2492, 440, 34,
...
Diagonal: a(n,n) = F(n+1).
First column: a(n,0) = F(2n+1) (A001519).
Row sums: Sum_{k=0..n} a(n,k) = F(3n+1) (A033887).
Alternated row sums: Sum_{k=0..n} (-1)^k * a(n,k) = 1.
Diagonal sums: Sum_{k=0..floor(n/2)} a(n-k,k) = A208481(n).
Alternated diagonal sums: Sum_{k=0..floor(n/2)} (-1)^k * a(n-k,k) = F(n+3)-1 (A000071).
Row square-sums: Sum_{k=0..n} a(n,k)^2 = A208588(n).
Central coefficients: a(2*n,n) = binomial(2n,n)*F(3n+1) (A208473), where F(n) are the Fibonacci numbers (A000045).
Mirror image of the triangle in A122070. - Philippe Deléham, Mar 13 2012
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
FORMULA
a(n,k) = Sum_{i=k..n} binomial(n,i)*binomial(i,k)*F(i+1).
a(n,k) = binomial(n,i) * Sum_{i=k..n} binomial(n-k,n-i)*F(i+1).
Explicit form: a(n,k) = binomial(n,k)*F(2*n-k+1).
G.f.: (1-x)/(1-3*x+x^2-x*y-x^2*y-x^2*y^2).
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).
T(n,k) = A122070(n,n-k). - Philippe Deléham, Mar 13 2012
EXAMPLE
From Philippe Deléham, Mar 13 2012: (Start)
(1, 1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
5, 6, 2, 0;
13, 24, 15, 3, 0;
34, 84, 78, 32, 5, 0;
89, 275, 340, 210, 65, 8, 0;
233, 864, 1335, 1100, 510, 126, 13, 0;
... (End)
MATHEMATICA
Flatten[Table[Sum[Binomial[n, i]Binomial[i, k]Fibonacci[i+1], {i, k, n}], {n, 0, 20}, {k, 0, n}]]
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 *)
PROG
(Maxima) create_list(binomial(n, k)*fib(2*n-k+1), n, 0, 20, k, 0, n);
(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
KEYWORD
nonn,tabl,easy
AUTHOR
Emanuele Munarini, Feb 29 2012
STATUS
approved