A146956 A functionally symmetric Polynomial as a triangle of coefficients: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 2)*Sum[(2^(m-1) + 2*m-2 )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 38, 40, 1, 1, 125, 106, 106, 125, 1, 1, 406, 303, 276, 303, 406, 1, 1, 1383, 917, 739, 739, 917, 1383, 1, 1, 4936, 2972, 2104, 1862, 2104, 2972, 4936, 1, 1, 18313, 10276, 6484, 4990, 4990, 6484, 10276, 18313, 1, 1, 69898

Offset

0,5

Comments

Row sums are:{1, 2, 6, 28, 120, 464, 1696, 6080, 21888, 80128, 299520}.

Links

Table of n, a(n) for n=0..56.

Formula

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

Example

{1}, {1, 1}, {1, 4, 1}, {1, 13, 13, 1}, {1, 40, 38, 40, 1}, {1, 125, 106, 106, 125, 1}, {1, 406, 303, 276, 303, 406, 1}, {1, 1383, 917, 739, 739, 917, 1383, 1}, {1, 4936, 2972, 2104, 1862, 2104, 2972, 4936, 1}, {1, 18313, 10276, 6484, 4990, 4990, 6484, 10276, 18313, 1}, {1, 69898, 37421, 21624, 14546, \12540, 14546, 21624, 37421, 69898, 1}

Mathematica

Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 2)*Sum[(2^(m-1) + 2*m-2 )*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: A157180 A179086 A255494 * A152613 A157153 A212801

Adjacent sequences: A146953 A146954 A146955 * A146957 A146958 A146959

Keyword

nonn

Author

Roger L. Bagula, Nov 03 2008

Status

approved