A341725 Triangle read by rows: coefficients in expansion of Asveld's polynomials p_j(x).

1, 3, 1, 13, 6, 1, 81, 39, 9, 1, 673, 324, 78, 12, 1, 6993, 3365, 810, 130, 15, 1, 87193, 41958, 10095, 1620, 195, 18, 1, 1268361, 610351, 146853, 23555, 2835, 273, 21, 1, 21086113, 10146888, 2441404, 391608, 47110, 4536, 364, 24, 1

Offset: 0

Author

N. J. A. Sloane, Mar 04 2021

Keywords

• nonn
• tabl

Examples

			Triangle begins:
      1,
      3,     1,
     13,     6,     1,
     81,    39,     9,    1,
    673,   324,    78,   12,   1,
   6993,  3365,   810,  130,  15,  1,
  87193, 41958, 10095, 1620, 195, 18, 1,
  ...
		

References

• Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 5.

Links

• P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.

Crossrefs

Cf. A000557, A341723, A341724.

Column 0 is A005923, column 1 is A341728.

Programs

Maple

egf:= k-> exp(x)*x^k / ((1-2*sinh(x))*k!):
A341725:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341725(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 04 2023

Formula

From Mélika Tebni, Sep 04 2023: (Start)

T(n,k) = binomial(n,k)*A005923(n-k).

E.g.f. of column k: exp(x)*x^k / ((1-2*sinh(x))*k!).

T(n,k) = Sum_{j=k..n} binomial(n,j)*A000557(n-j)*binomial(j,k).

Recurrence: T(n,0) = A005923(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)

Sum_{k=0..n} (-1)^k * T(n,k) = A000557(n). - Alois P. Heinz, Sep 04 2023

Extensions

More terms from Mélika Tebni, Sep 04 2023