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A357255 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.

%I A357255 #20 Mar 23 2025 20:53:14
%S A357255 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 A357255 -55,36,-13,2,9,-35,77,-105,91,-49,15,-2,10,-44,112,-182,196,-140,64,
%U A357255 -17,2,11,-54,156,-294,378,-336,204,-81,19,-2
%N A357255 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 A357255 n-th row sum = 1 for n >= 2.
%F A357255 T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.
%F A357255 T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.
%e A357255 First 7 rows:
%e A357255   2
%e A357255   3      -2
%e A357255   4      -5       2
%e A357255   5      -9       7     -2
%e A357255   6     -14      16     -9     2
%e A357255   7     -20      30    -25    11     -2
%e A357255   8     -27      50    -55    36    -13     2
%e A357255 Row 4 gives recurrence coefficients for the sequence
%e A357255 (r(k)) = (A002662(k)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
%e A357255 r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
%e A357255 with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
%e A357255 (Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)
%t A357255 Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]
%Y A357255 Cf. A029638, A029635.
%Y A357255 Cf. sequences generated by recurrences, by row, beginning with row 1: A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039.
%K A357255 tabl,sign
%O A357255 1,1
%A A357255 _Clark Kimberling_, Sep 24 2022