OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
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
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