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A251636
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Inverse of the Riordan array A251634: Riordan ((1-3*x)/(1-2*x), x/(1-2*x)).
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2
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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*z)/(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(T(n-1,k), k=1..n-1)) for n >= 1; T(0, 0) = 1.
Of course, T(1, 0) = 1 and T(n, 0) = -2^(n-1) for n >= 1.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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