A005921 -id:A005921 - cp's OEIS frontend

A005922 a(1)=1; a(n) = n!*Fibonacci(n+2), n > 1.

1, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320, 522547200, 9300614400, 180583603200, 3798482688000, 86044973414400, 2088355965696000, 54064489070592000, 1487129136869376000, 43312058119249920000

Offset: 1

N. J. A. Sloane

From solution to a difference equation.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.

Cf. A005921, A078700.

[1] cat [Factorial(n)*Fibonacci(n+2): n in [2..20]]; // Vincenzo Librandi, Jul 10 2012

Join[{1},Table[n!Fibonacci[n+2],{n,2,20}]] (* Harvey P. Dale, May 26 2024 *)

a(n)=if(n>1, n!*fibonacci(n+2), 1) \\ Charles R Greathouse IV, Jun 30 2017

More terms and better description from Vladeta Jovovic, Jan 23 2005

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

Original entry on oeis.org

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

Mélika Tebni, Sep 23 2023

			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,
  ...

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.

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

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);

T(n,k) = n!/k!*sum(j=k, n, fibonacci(j-k+1)*binomial(2,n-j)) \\ Winston de Greef, Oct 21 2023

T(n,k) = if(n == k, 1, n!/k!*fibonacci(n-k+3)) \\ Winston de Greef, Oct 21 2023

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).

cp's OEIS Frontend

A005922 a(1)=1; a(n) = n!*Fibonacci(n+2), n > 1.

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

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