A157526 Triangular sequence from coefficients of the polynomial recursion: p(x,n)=Sum[Binomial[n, m]*p[x, m]*p[x, n - m - 1], {m, 0, n - 1}].

1, 1, 1, 3, 3, 15, 18, 3, 105, 150, 45, 945, 1575, 675, 45, 10395, 19845, 11025, 1575, 135135, 291060, 198450, 44100, 1575, 2027025, 4864860, 3929310, 1190700, 99225, 34459425, 91216125, 85135050, 32744250, 4465125, 99225, 654729075, 1895268375

Offset

0,4

Comments

Row sums are:

{1, 2, 6, 36, 300, 3240, 42840, 670320, 12111120, 248119200, 5683154400,...}.

The first column is A001147:

{1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075,...}.

Links

Table of n, a(n) for n=0..36.

Formula

p(x,n)=Sum[Binomial[n, m]*p[x, m]*p[x, n - m - 1], {m, 0, n - 1}].

Example

{1},
{1, 1},
{3, 3},
{15, 18, 3},
{105, 150, 45},
{945, 1575, 675, 45},
{10395, 19845, 11025, 1575},
{135135, 291060, 198450, 44100, 1575},
{2027025, 4864860, 3929310, 1190700, 99225},
{34459425, 91216125, 85135050, 32744250, 4465125, 99225},
{654729075, 1895268375, 2006754750, 936485550, 180093375, 9823275}

Mathematica

p[x, 0] = 1;
p[x, 1] = x + 1;
p[x_, n_] := Sum[Binomial[n, m]*p[x, m]*p[x, n - m - 1], {m, 0, n - 1}];
Table[ExpandAll[p[x, n]], {n, 0, 10}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

A001147

Sequence in context: A232097 A353584 A110096 * A208229 A269956 A153512

Adjacent sequences: A157523 A157524 A157525 * A157527 A157528 A157529

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 02 2009

Status

approved