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

Author

Emanuele Munarini, Feb 29 2012

Keywords

• nonn
• tabl
• easy

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

Examples

			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)
		

Links

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

Crossrefs

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

Cf. A122070.

Programs

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 *)

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

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