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

0,2

Comments

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

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

Links

Table of n, a(n) for n=0..49.

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

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

Example

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

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 *)

Crossrefs

The leading column is A107841.

Cf. A103210, A107841.

Sequence in context: A136216 A121334 A126450 * A112333 A066868 A193900

Adjacent sequences: A235605 A235606 A235607 * A235609 A235610 A235611

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 23 2014

Extensions

More terms from Philippe Deléham, Jan 31 2014

Status

approved