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A341723 Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.

%I A341723 #33 Sep 11 2023 12:41:18
%S A341723 1,-1,1,5,-2,1,-31,15,-3,1,257,-124,30,-4,1,-2671,1285,-310,50,-5,1,
%T A341723 33305,-16026,3855,-620,75,-6,1,-484471,233135,-56091,8995,-1085,105,
%U A341723 -7,1,8054177,-3875768,932540,-149576,17990,-1736,140,-8,1
%N A341723 Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.
%C A341723 Conjectures from _Mélika Tebni_, Sep 09 2023: (Start)
%C A341723 For 0 < k < p and p prime, T(p,k) == 0 (mod p).
%C A341723 For 0 < k < n (k odd) and n = 2^m (m natural number), T(n,k) == 0 (mod n). (End)
%D A341723 Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 2.
%F A341723 From _Mélika Tebni_, Sep 09 2023: (Start)
%F A341723 E.g.f. of column k: exp(x)*x^k / ((1+2*sinh(x))*k!).
%F A341723 T(n,k) = (-1)^(n-k)*binomial(n,k)*A000556(n-k).
%F A341723 Recurrence: T(n,0) = (-1)^n*A000556(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)
%e A341723 Triangle begins:
%e A341723         1;
%e A341723        -1,        1;
%e A341723         5,       -2,      1;
%e A341723       -31,       15,     -3,       1;
%e A341723       257,     -124,     30,      -4,     1;
%e A341723     -2671,     1285,   -310,      50,    -5,     1;
%e A341723     33305,   -16026,   3855,    -620,    75,    -6,   1;
%e A341723   -484471,   233135, -56091,    8995, -1085,   105,  -7,  1;
%e A341723   8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 1;
%e A341723   ...
%p A341723 egf:= k-> exp(x)*x^k / ((1+2*sinh(x))*k!):
%p A341723 A341723:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A341723 seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # _Mélika Tebni_, Sep 09 2023
%p A341723 second Maple program:
%p A341723 A341723:= (n, k)-> (-1)^(n-k)*binomial(n, k)*add(j!*combinat[fibonacci](j+1)*Stirling2(n-k,j), j=0.. n-k):
%p A341723 seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # _Mélika Tebni_, Sep 09 2023
%Y A341723 Column 0 is a signed version of A000556, column 1 is A341726.
%Y A341723 Cf. A341724, A341725.
%K A341723 sign,tabl
%O A341723 0,4
%A A341723 _N. J. A. Sloane_, Mar 04 2021