A248156 Inverse Riordan triangle of A106513: Riordan ((1 - 2*x^2 )/(1 + x), x/(1+x)).

1, -2, 1, 1, -3, 1, 0, 4, -4, 1, -1, -4, 8, -5, 1, 2, 3, -12, 13, -6, 1, -3, -1, 15, -25, 19, -7, 1, 4, -2, -16, 40, -44, 26, -8, 1, -5, 6, 14, -56, 84, -70, 34, -9, 1, 6, -11, -8, 70, -140, 154, -104, 43, -10, 1, -7, 17, -3, -78, 210, -294, 258, -147, 53, -11, 1, 8, -24, 20, 75, -288, 504, -552, 405, -200, 64, -12, 1

Offset

0,2

Comments

Columns k=0..3 give A248157, A248158, A248159, A248160.

Row sums have o.g.f. (1 - 2*x)/(1 + x): [1, -1, repeat(-1, 1)].

Alternating row sums are (-1)^n*A083318(n).

Links

Table of n, a(n) for n=0..77.

Wolfdieter Lang, First 13 rows of the triangle

Formula

O.g.f. row polynomials R(n,x) = sum(T(n,k)*x^k, k=0..n): (1 - 2*z)/((1 + z)*(1 + (1-x)*z)).

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

The A-sequence is [1, -1], implying the recurrence T(n,k) = T(n-1,k-1) - T(n-1,k), n >= k > = 1. The Z-sequence is -[2, 3, 7, 17, 41, 99, 239, 577, 1393, ...] = A248161, implying the recurrence T(n,0) = sum(T(n-1,k)*Z(k),k=0..n-1). See the W. Lang link under A006232 for Riordan A- and Z-sequences.

The standard recurrence for the sequence for column k=0 is T(0,0) = 1 and T(n,0) = -2*T(n-1,0) - T(n-2,0), n >= 3, with T(1,0) = -2 and T(2,0) = 1.

Example

The triangle T(n,k) begins:
n\k  0   1   2   3    4    5    6    7    8    9
0:   1
1:  -2   1
2:   1  -3   1
3:   0   4  -4   1
4:  -1  -4   8  -5    1
5:   2   3 -12  13   -6    1
6:  -3  -1  15 -25   19   -7    1
7:   4  -2 -16  40  -44   26   -8    1
8:  -5   6  14 -56   84  -70   34   -9    1
9:   6 -11  -8  70 -140  154 -104   43  -10    1
...
For more rows see the link.
Recurrence from A-sequence:  -12 = T(5,2) = T(4,1) - T(4,2) = -4 - 8.
Recurrence from the Z-sequence:  2 = T(5,0) = -(2*(-1) + 3*(-4) + 7*8 + 17*(-5) + 41*1) = 2.
Standard recurrence for T(n,0):  0 = T(3,0) = -2*T(2,0) - T(1,0) = -2*1 -(-2).

Mathematica

T[n_, k_] := SeriesCoefficient[x^k*(1 - 2*x^2)/(1 + x)^(k + 2), {x, 0, n}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 09 2014 *)

Crossrefs

Cf. A106513, A083318, A248157, A248158, A248159, A248160, A248161.

Sequence in context: A226173 A212633 A202241 * A352780 A375706 A331368

Adjacent sequences: A248153 A248154 A248155 * A248157 A248158 A248159

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Oct 05 2014

Status

approved