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 A235608 Triangle read by rows: a non-Riordan array serving as a counter-example to a conjecture about Riordan arrays. 0

%I

%S 1,2,1,10,5,1,62,31,7,1,430,215,51,10,1,3194,1597,389,87,12,1,24850,

%T 12425,3077,740,117,15,1,199910,99955,25035,6305,1076,168,17,1,

%U 1649350,824675,208255,54150,9705,1700,208,20,1,13879538,6939769,1763473,469399,87048

%N Triangle read by rows: a non-Riordan array serving as a counter-example to a conjecture about Riordan arrays.

%C See Barry (2013), Example 3, for precise definition.

%C T(n,1) = T(n,0)/2 for n>0.- _Philippe Deléham_, Jan 31 2014

%H P. Barry, <a href="http://arxiv.org/abs/1312.0583">Embedding structures associated with Riordan arrays and moment matrices</a>, arXiv preprint arXiv:1312.0583, 2013. See Example 3.

%F G.f. for the column k (with leading zero omitted): f(x)^floor((k+2)/2))*g(x)^floor((k+1)/2)) with f(x) = (1+x-sqrt(1-10*x+x^2))/(6*x) and g(x) = (1-x-sqrt(1-10*x+x^2))/(4*x). - _Philippe Deléham_, Jan 31 2014

%e Triangle begins:

%e 1;

%e 2, 1;

%e 10, 5, 1;

%e 62, 31, 7, 1;

%e 430, 215, 51, 10, 1;

%e 3194, 1597, 389, 87, 12, 1;

%e 24850, 12425, 3077, 740, 117, 15, 1;

%e 199910, 99955, 25035, 6305, 1076, 168, 17, 1;

%e 1649350, 824675, 208255, 54150, 9705, 1700, 208, 20, 1;

%e 13879538, 6939769, 1763473, 469399, 87048, 16449, 2248, 274, 22, 1;

%e ... - Extended by _Philippe Deléham_, Jan 31 2014

%t f[x_] := (1+x-Sqrt[1-10*x+x^2])/(6*x); g[x_] := (1-x-Sqrt[1-10*x+x^2])/(4*x); t[n_, k_] := SeriesCoefficient[f[x]^Floor[(k+2)/2]*g[x]^Floor[(k+1)/2], {x, 0, n}]; Table[t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 31 2014, after _Philippe Deléham_ *)

%Y The leading column is A107841.

%Y Cf. A103210, A107841,

%K nonn,tabl

%O 0,2

%A _N. J. A. Sloane_, Jan 23 2014

%E More terms from _Philippe Deléham_, Jan 31 2014

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Last modified August 19 18:51 EDT 2019. Contains 326133 sequences. (Running on oeis4.)