A146955 A functionally symmetric Polynomial as a triangle of coefficients: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[(2^m + 2*m )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 22, 22, 22, 1, 1, 61, 54, 54, 61, 1, 1, 190, 143, 132, 143, 190, 1, 1, 647, 421, 339, 339, 421, 647, 1, 1, 2344, 1372, 952, 838, 952, 1372, 2344, 1, 1, 8841, 4836, 2964, 2238, 2238, 2964, 4836, 8841, 1, 1, 34186, 17965, 10104, 6610

Offset

0,5

Comments

Row sums are:{1, 2, 6, 20, 68, 232, 800, 2816, 10176, 37760, 143360}.

Links

Table of n, a(n) for n=0..59.

Formula

p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[(2^m + 2*m )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).

Conjecture: row sums are 2^n*(2^n+6-n+n^2)/8 for n>0. [From R. J. Mathar, Nov 30 2008]

Example

{1},
{1, 1},
{1, 4, 1},
{1, 9, 9, 1},
{1, 22, 22, 22, 1},
{1, 61, 54, 54, 61, 1},
{1, 190, 143, 132, 143, 190, 1},
{1, 647, 421, 339, 339, 421, 647, 1},
{1, 2344, 1372, 952, 838, 952, 1372, 2344, 1},
{1, 8841, 4836, 2964, 2238, 2238, 2964, 4836, 8841, 1},
{1, 34186, 17965, 10104, 6610, 5628, 6610, 10104, 17965, 34186, 1}

Mathematica

Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[(2^m + 2*m )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

Crossrefs

Sequence in context: A154982 A347972 A146767 * A155451 A220681 A189280

Adjacent sequences: A146952 A146953 A146954 * A146956 A146957 A146958

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 03 2008

Status

approved