A395640 - OEIS

%I #16 Jun 20 2026 22:51:51

%S 0,1,-1,4,-5,1,18,-26,9,-1,96,-154,71,-14,1,600,-1044,580,-155,20,-1,

%T 4320,-8028,5104,-1665,295,-27,1,35280,-69264,48860,-18424,4025,-511,

%U 35,-1,322560,-663696,509004,-214676,54649,-8624,826,-44,1,3265920,-6999840

%N Triangular array: T(n, k) is the coefficient of x^k in the polynomial - n!*L(n, x), where L(n, x) is the Lagrange interpolating polynomial that passes through (1,0) and (k,1) for k = 2..n+1.

%e First eight rows:

%e       0

%e       1      -1

%e       4      -5      1

%e      18     -26      9      -1

%e      96    -154     71     -14     1

%e     600   -1044    580    -155    20    -1

%e    4320   -8028   5104   -1665   295   -27   1

%e   35280  -69264  48860  -18424  4025  -511  35   -1

%e L(4, x) = -4 + (77 x)/12 - (71 x^2)/24 + (7 x^3)/12 - x^4/24,

%e so 4!*L (4, x) = -96 + 154 x - 71 x^2 + 14 x^3 - x^4, whence row 4 of the array is (96, -154, 71, -14, 1).

%t t[0] = {0}; t[n_] := t[n] = Join[t[n - 1], {1}];

%t p[n_, x_] := Expand[InterpolatingPolynomial[t[n], x]];

%t Column[Join[{{0}}, Table[CoefficientList[-n!*p[n, x], x], {n, 1, 10}]]]

%t Join[{0}, Flatten[Table[CoefficientList[-n!*p[n, x], x], {n, 0, 10}]]]

%t Labeled[Plot[{p[1, x], p[2, x], p[3, x], p[4, x], p[5, x]}, {x, -1, 7}], "Plot of p[n,x] for n=1..5."]

%Y Cf. A001563 (column 1), A001701 (column 2), A001706 (column 3), A001797 (column 4), A395641, A395642, A004442, A125577, A000531, A000346, A396722, A396723, A396724.

%K tabl,sign

%O 0,4

%A _Clark Kimberling_, Jun 03 2026

%E Name edited by _Peter Munn_, Jun 16 2026