A366133 Triangle read by rows: coefficients in expansion of another Asveld's polynomials Pi_j(x).

1, 1, 1, 3, 2, 1, 8, 9, 3, 1, 50, 32, 18, 4, 1, 214, 250, 80, 30, 5, 1, 2086, 1284, 750, 160, 45, 6, 1, 11976, 14602, 4494, 1750, 280, 63, 7, 1, 162816, 95808, 58408, 11984, 3500, 448, 84, 8, 1, 1143576, 1465344, 431136, 175224, 26964, 6300, 672, 108, 9, 1, 20472504, 11435760, 7326720, 1437120, 438060, 53928, 10500, 960, 135, 10, 1

Offset: 0

Author

Mélika Tebni, Sep 30 2023

Keywords

• sign
• tabl

Comments

First negative term is T(35,0) = -230450728485788167742674544892530875760640.

Conjectures: For 0 < k < p and p prime, T(p,k) == 0 (mod p).

For 0 < k < n (k odd) and n = 2^m (m natural number), T(n,k) == 0 (mod n).

Examples

			Triangle begins:
      1,
      1,     1,
      3,     2,    1,
      8,     9,    3,    1,
     50,    32,   18,    4,   1,
    214,   250,   80,   30,   5,  1,
   2086,  1284,  750,  160,  45,  6,  1,
  11976, 14602, 4494, 1750, 280, 63,  7,  1,
  ...
		

Links

• P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous part, Fib. Quart. 27 (1989), 303-309. See Table 2 p. 308.

Crossrefs

Cf. A000045, A005444 (col 0), A005445, A039948, A048994, A305923 (row sums).

Programs

Maple

T := (n, k) -> binomial(n,k)*add(j!*combinat[fibonacci](j+1)*Stirling1(n-k,j), j=0 .. n-k): seq(print(seq(T(n, k), k = 0 .. n)), n=0 .. 9);
# second Maple program:
T := (n, k) -> add(Stirling2(j, k)/j!*add(i!*combinat[fibonacci](i-j+1)*Stirling1(n, i), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);

Formula

T(n,k) = binomial(n,k)*A005444(n-k).

Sum_{k=1..n} (-1)^(k-1)*(k-1)!*T(n, k) = A005445(n).

E.g.f. of column k: x^k / ((1-log(1+x)-log(1+x)^2)*k!), k >= 0.

Recurrence: T(n,0) = A005444(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)*A039948(i,j)).