A185384 A binomial transform of Fibonacci numbers.

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

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

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

Crossrefs

Cf. A000045, A001519, A033887, A208481, A000071, A208588, A208473.

Cf. A122070.

Sequence in context: A231774 A209170 A231732 * A274728 A062991 A234950

Adjacent sequences: A185381 A185382 A185383 * A185385 A185386 A185387

Keyword

nonn,tabl,easy

Author

Emanuele Munarini, Feb 29 2012

Status

approved