cp's OEIS Frontend

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

%I A341725 #26 Jan 05 2025 19:51:41
%S A341725 1,3,1,13,6,1,81,39,9,1,673,324,78,12,1,6993,3365,810,130,15,1,87193,
%T A341725 41958,10095,1620,195,18,1,1268361,610351,146853,23555,2835,273,21,1,
%U A341725 21086113,10146888,2441404,391608,47110,4536,364,24,1
%N A341725 Triangle read by rows: coefficients in expansion of Asveld's polynomials p_j(x).
%D A341725 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 A341725 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 A341725 From _Mélika Tebni_, Sep 04 2023: (Start)
%F A341725 T(n,k) = binomial(n,k)*A005923(n-k).
%F A341725 E.g.f. of column k: exp(x)*x^k / ((1-2*sinh(x))*k!).
%F A341725 T(n,k) = Sum_{j=k..n} binomial(n,j)*A000557(n-j)*binomial(j,k).
%F A341725 Recurrence: T(n,0) = A005923(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)
%F A341725 Sum_{k=0..n} (-1)^k * T(n,k) = A000557(n). - _Alois P. Heinz_, Sep 04 2023
%e A341725 Triangle begins:
%e A341725       1,
%e A341725       3,     1,
%e A341725      13,     6,     1,
%e A341725      81,    39,     9,    1,
%e A341725     673,   324,    78,   12,   1,
%e A341725    6993,  3365,   810,  130,  15,  1,
%e A341725   87193, 41958, 10095, 1620, 195, 18, 1,
%e A341725   ...
%p A341725 egf:= k-> exp(x)*x^k / ((1-2*sinh(x))*k!):
%p A341725 A341725:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A341725 seq(print(seq(A341725(n, k), k=0..n)), n=0..8); # _Mélika Tebni_, Sep 04 2023
%Y A341725 Cf. A000557, A341723, A341724.
%Y A341725 Column 0 is A005923, column 1 is A341728.
%K A341725 nonn,tabl
%O A341725 0,2
%A A341725 _N. J. A. Sloane_, Mar 04 2021
%E A341725 More terms from _Mélika Tebni_, Sep 04 2023