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

0,3

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.

Links

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

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).

Example

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;

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 *)

Prog

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

Crossrefs

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

Sequence in context: A019749 A209773 A209767 * A181661 A144160 A368228

Adjacent sequences: A122067 A122068 A122069 * A122071 A122072 A122073

Keyword

nonn,tabl

Author

Philippe Deléham, Oct 15 2006, Mar 13 2012

Extensions

Corrected and extended by Philippe Deléham, Mar 13 2012

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

Status

approved