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.

2, 3, -2, 4, -5, 2, 5, -9, 7, -2, 6, -14, 16, -9, 2, 7, -20, 30, -25, 11, -2, 8, -27, 50, -55, 36, -13, 2, 9, -35, 77, -105, 91, -49, 15, -2, 10, -44, 112, -182, 196, -140, 64, -17, 2, 11, -54, 156, -294, 378, -336, 204, -81, 19, -2

Offset: 1

Author

Clark Kimberling, Sep 24 2022

Keywords

• tabl
• sign

Comments

n-th row sum = 1 for n >= 2.

Examples

			First 7 rows:
  2
  3      -2
  4      -5       2
  5      -9       7     -2
  6     -14      16     -9     2
  7     -20      30    -25    11     -2
  8     -27      50    -55    36    -13     2
Row 4 gives recurrence coefficients for the sequence
(r(k)) = (A002662(k)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
(Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)
		

Crossrefs

Cf. A029638, A029635.

Cf. sequences generated by recurrences, by row, beginning with row 1: A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039.

Programs

Mathematica

Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]

Formula

T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.

T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.