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

%I A235608 #31 Feb 25 2025 03:53:37
%S A235608 1,2,1,10,5,1,62,31,7,1,430,215,51,10,1,3194,1597,389,87,12,1,24850,
%T A235608 12425,3077,740,117,15,1,199910,99955,25035,6305,1076,168,17,1,
%U A235608 1649350,824675,208255,54150,9705,1700,208,20,1,13879538,6939769,1763473,469399,87048
%N A235608 Triangle read by rows: a non-Riordan array serving as a counterexample to a conjecture about Riordan arrays.
%C A235608 See Barry (2013), Example 3, for precise definition.
%C A235608 T(n,1) = T(n,0)/2 for n > 0. - _Philippe Deléham_, Jan 31 2014
%H A235608 Paul Barry, <a href="http://arxiv.org/abs/1312.0583">Embedding structures associated with Riordan arrays and moment matrices</a>, arXiv preprint arXiv:1312.0583 [math.CO], 2013. See Example 3.
%F A235608 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 A235608 Triangle begins:
%e A235608          1;
%e A235608          2,       1;
%e A235608         10,       5,       1;
%e A235608         62,      31,       7,      1;
%e A235608        430,     215,      51,     10,     1;
%e A235608       3194,    1597,     389,     87,    12,    1;
%e A235608      24850,   12425,    3077,    740,   117,    15,    1;
%e A235608     199910,   99955,   25035,   6305,  1076,   168,   17,   1;
%e A235608    1649350,  824675,  208255,  54150,  9705,  1700,  208,  20,  1;
%e A235608   13879538, 6939769, 1763473, 469399, 87048, 16449, 2248, 274, 22, 1;
%e A235608 ... - Extended by _Philippe Deléham_, Jan 31 2014
%t A235608 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 A235608 The leading column is A107841.
%Y A235608 Cf. A103210, A107841.
%K A235608 nonn,tabl
%O A235608 0,2
%A A235608 _N. J. A. Sloane_, Jan 23 2014
%E A235608 More terms from _Philippe Deléham_, Jan 31 2014