OFFSET
1,4
COMMENTS
Row sums are: {1, 2, 7, 34, 261, 3081, 57279, 1676641, 77766297, 5728225636, 671925730146, ...}.
REFERENCES
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page86
LINKS
G. C. Greubel, Rows n = 1..100 of triangle, flattened
FORMULA
With f(n) = Fibonacci(n)*f(n-1) then the triangle is formed by L(n, k) = binomial(n-1, k-1)*(f(n)/f(k)).
With f(n) = Product_{j=1..n} Fibonacci(j) then the triangle is formed by T(n, k) = binomial(n-1, k-1)*(f(n)/f(k)). - G. C. Greubel, May 15 2019
EXAMPLE
Triangle begins as:
1;
1, 1;
2, 4, 1;
6, 18, 9, 1;
30, 120, 90, 20, 1;
240, 1200, 1200, 400, 40, 1;
3120, 18720, 23400, 10400, 1560, 78, 1;
65520, 458640, 687960, 382200, 76440, 5733, 147, 1;
MATHEMATICA
f[n_]:= Product[Fibonacci[j], {j, 1, n}]; Table[Binomial[n-1, k-1]* f[n]/f[k], {n, 1, 12}, {k, 1, n}]//Flatten (* G. C. Greubel, May 15 2019 *)
PROG
(PARI)
{f(n) = prod(j=1, n, fibonacci(j))};
{T(n, k) = binomial(n-1, k-1)*(f(n)/f(k))};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 15 2019
(Magma)
f:= func< n | (&*[Fibonacci(j): j in [1..n]]) >;
[[Binomial(n-1, k-1)*(f(n)/f(k)): k in [1..n]]: n in [1..12]]; // G. C. Greubel, May 15 2019
(Sage)
def f(n): return product(fibonacci(j) for j in (1..n))
[[binomial(n-1, k-1)*(f(n)/f(k)) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 15 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 22 2008
EXTENSIONS
Edited by G. C. Greubel, May 15 2019
STATUS
approved