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

1, 2, 1, 10, 5, 1, 62, 31, 7, 1, 430, 215, 51, 10, 1, 3194, 1597, 389, 87, 12, 1, 24850, 12425, 3077, 740, 117, 15, 1, 199910, 99955, 25035, 6305, 1076, 168, 17, 1, 1649350, 824675, 208255, 54150, 9705, 1700, 208, 20, 1, 13879538, 6939769, 1763473, 469399, 87048

Offset: 0

Author

N. J. A. Sloane, Jan 23 2014

Keywords

• nonn
• tabl

Comments

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

T(n,1) = T(n,0)/2 for n > 0. - Philippe Deléham, Jan 31 2014

Examples

			Triangle begins:
         1;
         2,       1;
        10,       5,       1;
        62,      31,       7,      1;
       430,     215,      51,     10,     1;
      3194,    1597,     389,     87,    12,    1;
     24850,   12425,    3077,    740,   117,    15,    1;
    199910,   99955,   25035,   6305,  1076,   168,   17,   1;
   1649350,  824675,  208255,  54150,  9705,  1700,  208,  20,  1;
  13879538, 6939769, 1763473, 469399, 87048, 16449, 2248, 274, 22, 1;
... - Extended by _Philippe Deléham_, Jan 31 2014
		

Links

• Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013. See Example 3.

Crossrefs

The leading column is A107841.

Cf. A103210, A107841.

Programs

Mathematica

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 *)

Formula

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

Extensions

More terms from Philippe Deléham, Jan 31 2014