A039948 A triangle related to A000045 (Fibonacci numbers).

1, 1, 1, 4, 2, 1, 18, 12, 3, 1, 120, 72, 24, 4, 1, 960, 600, 180, 40, 5, 1, 9360, 5760, 1800, 360, 60, 6, 1, 105840, 65520, 20160, 4200, 630, 84, 7, 1, 1370880, 846720, 262080, 53760, 8400, 1008, 112, 8, 1, 19958400, 12337920, 3810240, 786240, 120960, 15120, 1512, 144, 9, 1

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• tabl

Examples

			Triangle begins :
    1;
    1,   1;
    4,   2,   1;
   18,  12,   3,  1;
  120,  72,  24,  4, 1;
  960, 600, 180, 40, 5, 1;
... - _Philippe Deléham_, Nov 08 2011
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened
• P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Crossrefs

Cf. A000045, A000142, A005442, A005443, A110313 (row sums).

Diagonals include: A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288, A010965, A010966.

Programs

Magma

[(Factorial(n)/Factorial(k))*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2022

Mathematica

T[n_,k_]:= (n!/k!)*Fibonacci[n-k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2022 *)

SageMath

def A039948(n, k): return factorial(n-k)*binomial(n,k)*fibonacci(n-k+1)
flatten([[A039948(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2022

Formula

T(n, m) = n!*Fibonacci(n-m+1)/m!, n >= m >= 0.

T(n, 0) = A005442(n).

T(n, 1) = A005443(n).

E.g.f. for column m: x^m/(m!*(1-x-x^2)), m >= 0.

From G. C. Greubel, Nov 20 2022: (Start)

T(n, n-1) = A000027(n).

T(n, n-2) = 4*A000217(n-1), n >= 2.

T(n, n-3) = 18*A000292(n-2), n >= 3.

T(n, n-4) = 5! * A000332(n), n >= 4.

T(n, n-5) = 8 * 5! * A000389(n), n >= 5.

T(n, n-6) = 13 * 6! * A000579(n), n >= 6.

T(n, n-7) = 21 * 7! * A000580(n), n >= 7.

T(n, n-8) = 34 * 8! * A000581(n), n >= 8.

T(n, n-9) = 55 * 9! * A000582(n), n >= 9.

T(n, n-10) = 89 * 10! * A001287(n), n >= 10.

T(n, n-11) = 12 * 12! * A001288(n), n >= 11.

T(n, n-12) = 233 * 12! * A010965(n), n >= 12.

T(n, n-13) = 89 * 13! * A010966(n), n >= 13.

Sum_{k=0..n} T(n, k) = A110313(n). (End)