A341728 -id:A341728 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 04 2021

			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,
  ...

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.

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

Cf. A000557, A341723, A341724.

Column 0 is A005923, column 1 is A341728.

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

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

More terms from Mélika Tebni, Sep 04 2023

A341727 Column 1 of A341724.

Original entry on oeis.org

1, -4, 24, -200, 2080, -25932, 377216, -6271120, 117287424, -2437334420, 55714920448, -1389365372760, 37533886128128, -1091981490075868, 34038486791454720, -1131758947520249120, 39982188936149204992, -1495556350390047594276, 59050025742595595436032

Offset: 1

N. J. A. Sloane, Mar 04 2021

sign

Cf. A000556, A000557, A341724 ,A341726, A341728.

A341727 := n -> (-1)^(n-1)*n*add(k!*combinat[fibonacci](k+2)*Stirling2(n-1, k), k=0..n-1):seq(A341727(n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023
# E.g.f. Maple program:
A341727 := series(x / (1 + 2*sinh(x)), x = 0, 20):
seq(n!*coeff(A341727, x, n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023

From Mélika Tebni, Sep 04 2023: (Start)
E.g.f.: x / (1 + 2*sinh(x)).
a(n) = (-1)^(n-1)*n*A000557(n-1).
a(n) = (-1)^(n-1)*Sum_{k=0..n} A000556(k)*(n-k)*binomial(n, k). (End)

More terms from Mélika Tebni, Sep 04 2023

cp's OEIS Frontend

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

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