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

0,4

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.

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)

Example

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

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 *)

Prog

(Magma) [(Factorial(n)/Factorial(k))*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // 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

Crossrefs

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

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

Sequence in context: A256039 A152391 A144088 * A111536 A111559 A224798

Adjacent sequences: A039945 A039946 A039947 * A039949 A039950 A039951

Keyword

nonn,tabl

Author

Wolfdieter Lang

Status

approved