%I #17 Jul 07 2024 20:52:33
%S 2,3,-2,4,-5,2,5,-9,7,-2,6,-14,16,-9,2,7,-20,30,-25,11,-2,8,-27,50,
%T -55,36,-13,2,9,-35,77,-105,91,-49,15,-2,10,-44,112,-182,196,-140,64,
%U -17,2,11,-54,156,-294,378,-336,204,-81,19,-2
%N Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.
%C n-th row sum = 1 for n >= 2.
%F T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.
%F T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.
%e First 7 rows:
%e 2
%e 3 -2
%e 4 -5 2
%e 5 -9 7 -2
%e 6 -14 16 -9 2
%e 7 -20 30 -25 11 -2
%e 8 -27 50 -55 36 -13 2
%e Row 4 gives recurrence coefficients for the sequence
%e (r(k)) = (A002662)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
%e r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
%e with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
%e (Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)
%t Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]
%Y Cf. A029638, A029635.
%Y Cf. sequences generated by recurrences, by row, beginning with row 1: A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039.
%K tabl,sign
%O 1,1
%A _Clark Kimberling_, Sep 24 2022