A122070 Triangle given by T(n,k) = Fibonacci(n+k+1)*binomial(n,k) for 0<=k<=n.

1, 1, 2, 2, 6, 5, 3, 15, 24, 13, 5, 32, 78, 84, 34, 8, 65, 210, 340, 275, 89, 13, 126, 510, 1100, 1335, 864, 233, 21, 238, 1155, 3115, 5040, 4893, 2639, 610, 34, 440, 2492, 8064, 16310, 21112, 17080, 7896, 1597, 55, 801, 5184, 19572, 47502, 76860, 82908, 57492, 23256, 4181

Offset: 0

Author

Philippe Deléham, Oct 15 2006, Mar 13 2012

Keywords

• nonn
• tabl

Comments

Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Mirror image of the triangle in A185384.

Examples

			Triangle begins:
   1;
   1,   2;
   2,   6,   5;
   3,  15,  24,   13;
   5,  32,  78,   84,   34;
   8,  65, 210,  340,  275,  89;
  13, 126, 510, 1100, 1335, 864, 233;
(0, 1, 1, -1, 0, 0, ...) DELTA (1, 1, 1, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,   2;
  0,  2,   6,   5;
  0,  3,  15,  24,   13;
  0,  5,  32,  78,   84,   34;
  0,  8,  65, 210,  340,  275,  89;
  0, 13, 126, 510, 1100, 1335, 864, 233;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000045, A001519, A033887, A033889, A185384.

Programs

GAP

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n+ k+1) ))); # G. C. Greubel, Oct 02 2019

Magma

[Binomial(n,k)*Fibonacci(n+k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019

Maple

with(combinat): seq(seq(binomial(n,k)*fibonacci(n+k+1), k=0..n), n=0..10); # G. C. Greubel, Oct 02 2019

Mathematica

Table[Fibonacci[n+k+1]*Binomial[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)

PARI

T(n,k) = binomial(n,k)*fibonacci(n+k+1);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 02 2019

Sage

[[binomial(n,k)*fibonacci(n+k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 02 2019

Formula

T(n,k) = A000045(n+k+1)*A007318(n,k) .

T(n,n) = Fibonacci(2*n+1) = A001519(n+1) .

Sum_{k=0..n} T(n,k) = Fibonacci(3*n+1) = A033887(n) .

Sum_{k=0..n}(-1)^k*T(n,k) = (-1)^n = A033999(n) .

Sum_{k=0..floor(n/2)} T(n-k,k) = (Fibonacci(n+1))^2 = A007598(n+1).

Sum_{k=0..n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n).

Sum_{k=0..n} T(n,k)^2 = A208588(n).

G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2).

T(n,k) = T(n-1,k) + 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.

T(n,k) = A185384(n,n-k).

T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n).

Extensions

Corrected and extended by Philippe Deléham, Mar 13 2012

Term a(50) corrected by G. C. Greubel, Oct 02 2019