OFFSET
0,2
LINKS
Winston de Greef, Table of n, a(n) for n = 0..11324
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.
FORMULA
E.g.f. of column k: (1+x)^2*x^k / ((1-x-x^2)*k!), k >= 0.
T(n,n) = 1 and T(n,k) = n!/k!*Fibonacci(n-k+3), n > k >= 0.
T(n,k) = n!/k!*Sum_{j=k..n} Fibonacci(j-k+1)*binomial(2,n-j).
T(n,k) = n!/k!*Sum_{j=k..n} (Fibonacci(j-k)+(-1)^(j-k))*binomial(3,n-j).
Recurrence: T(n,0) = A005921(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A341725(i,j)).
Sum_{j=k..n} (-1)^(n-j)*(n-j+1)!*binomial(n,j)*T(j,k) = A039948(n,k).
EXAMPLE
Triangle begins:
1,
3, 1,
10, 6, 1,
48, 30, 9, 1,
312, 192, 60, 12, 1,
2520, 1560, 480, 100, 15, 1,
24480, 15120, 4680, 960, 150, 18, 1,
...
MAPLE
T := proc(n, k) option remember; if k = n then 1 else (n!/k!*combinat[fibonacci](n-k+3)) fi end: seq(print(seq(T(n, k), k = 0..n)), n=0..9);
# second Maple program:
T := (n, k) -> add(Stirling2(j, k)*add(Stirling1(n, i)*A341725(i, j), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);
PROG
(PARI) T(n, k) = n!/k!*sum(j=k, n, fibonacci(j-k+1)*binomial(2, n-j)) \\ Winston de Greef, Oct 21 2023
(PARI) T(n, k) = if(n == k, 1, n!/k!*fibonacci(n-k+3)) \\ Winston de Greef, Oct 21 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mélika Tebni, Sep 23 2023
STATUS
approved