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Triangle read by rows: coefficients in expansion of Asveld's polynomials p_j(x).
6

%I #26 Jan 05 2025 19:51:41

%S 1,3,1,13,6,1,81,39,9,1,673,324,78,12,1,6993,3365,810,130,15,1,87193,

%T 41958,10095,1620,195,18,1,1268361,610351,146853,23555,2835,273,21,1,

%U 21086113,10146888,2441404,391608,47110,4536,364,24,1

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

%D 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.

%H P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/asveld.pdf">A family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 81-83.

%F From _Mélika Tebni_, Sep 04 2023: (Start)

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

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

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

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

%F Sum_{k=0..n} (-1)^k * T(n,k) = A000557(n). - _Alois P. Heinz_, Sep 04 2023

%e Triangle begins:

%e 1,

%e 3, 1,

%e 13, 6, 1,

%e 81, 39, 9, 1,

%e 673, 324, 78, 12, 1,

%e 6993, 3365, 810, 130, 15, 1,

%e 87193, 41958, 10095, 1620, 195, 18, 1,

%e ...

%p egf:= k-> exp(x)*x^k / ((1-2*sinh(x))*k!):

%p A341725:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):

%p seq(print(seq(A341725(n, k), k=0..n)), n=0..8); # _Mélika Tebni_, Sep 04 2023

%Y Cf. A000557, A341723, A341724.

%Y Column 0 is A005923, column 1 is A341728.

%K nonn,tabl

%O 0,2

%A _N. J. A. Sloane_, Mar 04 2021

%E More terms from _Mélika Tebni_, Sep 04 2023