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A159856
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Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.
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1
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1, 2, -1, 3, -4, 1, 4, -11, 6, -1, 5, -26, 23, -8, 1, 6, -57, 72, -39, 10, -1, 7, -120, 201, -150, 59, -12, 1, 8, -247, 522, -501, 268, -83, 14, -1, 9, -502, 1291, -1524, 1037, -434, 111, -16, 1, 10, -1013, 3084, -4339, 3598, -1905, 656, -143, 18, -1
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OFFSET
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0,2
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COMMENTS
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A Riordan array - see the Luzon references.
The second column is A000295 signed. - Michel Marcus, Feb 14 2014
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LINKS
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Table of n, a(n) for n=0..54.
Ana Luzón, Iterative Processes Related to Riordan Arrays: The Reciprocation and the Inversion of Power Series, arXiv:0907.2328 [math.CO]; Discrete Math., 310 (2010), 3607-3618.
Ana Luzón and Manuel A. Morón, Riordan matrices in the reciprocation of quadratic polynomials, Linear Algebra Appl. 430 (2009), no. 8-9, 22542270.
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FORMULA
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From R. J. Mathar, May 31 2009: (Start)
Sum_{k=0..n} T(n,k) = A080956(n).
Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)
T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k). - Vladimir Kruchinin, Nov 22 2016
G.f.: (1-2*x)/(1-x)^2/(1-2*x+y*x). - Vladimir Kruchinin, Nov 22 2016
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EXAMPLE
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Triangle begins
1;
2, -1;
3, -4, 1;
4, -11, 6, -1;
5, -26, 23, -8, 1;
6, -57, 72, -39, 10, -1;
7, -120, 201, -150, 59, -12, 1;
...
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MATHEMATICA
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With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
+y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)
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PROG
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(Maxima)
T(n, k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k, x, 0, 15), x, n); /* Vladimir Kruchinin, Nov 22 2016 */
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CROSSREFS
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Cf. A000295, A047926, A080956, A181690.
Sequence in context: A247239 A198060 A327084 * A137649 A180915 A240783
Adjacent sequences: A159853 A159854 A159855 * A159857 A159858 A159859
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KEYWORD
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easy,sign,tabl
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AUTHOR
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Philippe Deléham, Apr 24 2009
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EXTENSIONS
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a(41) corrected by Georg Fischer, Feb 18 2020
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STATUS
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approved
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