A365962 Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).

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, 277200, 171360, 52920, 10920, 1680, 210, 21, 1, 3588480, 2217600, 685440, 141120, 21840, 2688, 280, 24, 1, 52254720, 32296320, 9979200, 2056320, 317520, 39312, 4032, 360, 27, 1

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

Cf. A000045, A005921 (col 0), A005922 (col 1), A039948, A341725.

Sequence in context: A091965 A171568 A107056 * A337273 A376787 A116384

Adjacent sequences: A365959 A365960 A365961 * A365963 A365964 A365965

Keyword

nonn,tabl

Author

Mélika Tebni, Sep 23 2023

Status

approved