A251636 Inverse of the Riordan array A251634: Riordan ((1-3*x)/(1-2*x), x/(1-2*x)).

1, -1, 1, -2, 1, 1, -4, 0, 3, 1, -8, -4, 6, 5, 1, -16, -16, 8, 16, 7, 1, -32, -48, 0, 40, 30, 9, 1, -64, -128, -48, 80, 100, 48, 11, 1, -128, -320, -224, 112, 280, 196, 70, 13, 1, -256, -768, -768, 0, 672, 672, 336, 96, 15, 1, -512, -1792, -2304, -768, 1344, 2016, 1344, 528, 126, 17, 1

Offset: 0

Author

Wolfdieter Lang, Jan 11 2015

Keywords

• sign
• tabl
• easy

Comments

For Riordan arrays and references see the link "Sheffer a- and z-sequences", first part, in A006232.

The row sums have o.g.f. 1, that is the sequence is 1,repeat(0,) = A000007.

The alternating row sums have o.g.f. (1 - 3*x)/(1-x), that is the sequence is 1,repeat(-2,).

The A-sequence of this Riordan array is 1,2,repeat(0,) leading to the recurrence for T(n, k) for n >= k >= 1 given in the formula section. The Z-sequence of this Riordan array has o.g.f. -(1 + 2*x)/(10x), that is the sequence is -1,repeat(-3,), leading to the recurrence for T(n, 0) for n >= 1 given in the formula section. For A- and Z-sequences see the above mentioned link.

Examples

			The triangle T(n, k) begins:
  n\k    0     1     2    3    4    5    6   7   8   9
  O:     1
  1:    -1     1
  2:    -2     1     1
  3:    -4     0     3    1
  4:    -8    -4     6    5    1
  5:   -16   -16     8   16    7    1
  6:   -32   -48     0   40   30    9    1
  7:   -64  -128   -48   80  100   48   11   1
  8:  -128  -320  -224  112  280  196   70  13   1
  9:  -256  -768  -768    0  672  672  336  96  15   1
  ...
See the link for eleven rows.
Recurrence from the A-sequence: T(3, 1) = T(2, 0) + 2*T(2, 1) = -2 + 2*1 = 0.
Recurrence from the Z-sequence: T(3,0) = -(T(2,0) + 3*(T(2,1) + T(2,2))) = -(-2 + 3*(1 + 1)) = -4.
		

Links

• Wolfdieter Lang, First eleven rows of the triangle.

Crossrefs

Cf. A000007, A251634.

Formula

O.g.f. for row polynomials P(n,x) = sum(T(n, k),k=0..n) is G(z,x) = (1 - 3*z)/(1 - (2 + x)*z) (Riordan property).

O.g.f. column k: ((1 - 3*x)/(1 - 2*x))*(x/(1 - 2*x))^k, k >= 0.

Recurrence from the A-sequence (see comment above):

T(n, k) = T(n-1, k-1) + 2*T(n-1, k), for n >= k >= 1; T(0, 0) = 1; T(n, k) = 0 if n < k.

Recurrence from the Z-sequence (see comment above):

T(n, 0) = -(T(n-1, 0) + 3*Sum_{k=1..n-1} T(n-1, k)) for n >= 1; T(0, 0) = 1.

Of course, T(0, 0) = 1 and T(n, 0) = -2^(n-1) for n >= 1.