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

%I A366133 #15 Jan 05 2025 19:51:42
%S A366133 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,
%T A366133 45,6,1,11976,14602,4494,1750,280,63,7,1,162816,95808,58408,11984,
%U A366133 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
%N A366133 Triangle read by rows: coefficients in expansion of another Asveld's polynomials Pi_j(x).
%C A366133 First negative term is T(35,0) = -230450728485788167742674544892530875760640.
%C A366133 Conjectures: For 0 < k < p and p prime, T(p,k) == 0 (mod p).
%C A366133 For 0 < k < n (k odd) and n = 2^m (m natural number), T(n,k) == 0 (mod n).
%H A366133 P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-4/asveld.pdf">Fibonacci-like differential equations with a polynomial nonhomogeneous part</a>, Fib. Quart. 27 (1989), 303-309. See Table 2 p. 308.
%F A366133 T(n,k) = binomial(n,k)*A005444(n-k).
%F A366133 Sum_{k=1..n} (-1)^(k-1)*(k-1)!*T(n, k) = A005445(n).
%F A366133 E.g.f. of column k: x^k / ((1-log(1+x)-log(1+x)^2)*k!), k >= 0.
%F A366133 Recurrence: T(n,0) = A005444(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
%F A366133 T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A039948(i,j)).
%e A366133 Triangle begins:
%e A366133       1,
%e A366133       1,     1,
%e A366133       3,     2,    1,
%e A366133       8,     9,    3,    1,
%e A366133      50,    32,   18,    4,   1,
%e A366133     214,   250,   80,   30,   5,  1,
%e A366133    2086,  1284,  750,  160,  45,  6,  1,
%e A366133   11976, 14602, 4494, 1750, 280, 63,  7,  1,
%e A366133   ...
%p A366133 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);
%p A366133 # second Maple program:
%p A366133 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);
%Y A366133 Cf. A000045, A005444 (col 0), A005445, A039948, A048994, A305923 (row sums).
%K A366133 sign,tabl
%O A366133 0,4
%A A366133 _Mélika Tebni_, Sep 30 2023