A146900 Symmetrical polynomial: t0(n,m)=If[Mod[2*Binomial[n, m], 2] - Mod[Binomial[n, m], 2] == 0, Binomial[n, m]/2, Binomial[n, m] + 1]; p(x,n)=If[n == 0, 1, (x + 1)^n + Sum[t0(n,m)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/2].

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 6, 9, 6, 1, 1, 11, 15, 15, 11, 1, 1, 9, 31, 30, 31, 9, 1, 1, 15, 43, 71, 71, 43, 15, 1, 1, 12, 42, 84, 105, 84, 42, 12, 1, 1, 19, 54, 126, 189, 189, 126, 54, 19, 1, 1, 15, 91, 180, 315, 378, 315, 180, 91, 15, 1

Offset

0,5

Comments

Row sums are:{1, 2, 5, 16, 23, 54, 112, 260, 383, 778, 1582}.

Links

Table of n, a(n) for n=0..65.

Formula

t0(n,m)=If[Mod[2*Binomial[n, m], 2] - Mod[Binomial[n, m], 2] == 0, Binomial[n, m]/2, Binomial[n, m] + 1]; p(x,n)=If[n == 0, 1, (x + 1)^n + Sum[t0(n,m)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/2]; t(n,m)=coefficients(p(x,n)).

Example

{1}, {1, 1}, {1, 3, 1}, {1, 7, 7, 1}, {1, 6, 9, 6, 1}, {1, 11, 15, 15, 11, 1}, {1, 9, 31, 30, 31, 9, 1}, {1, 15, 43, 71, 71, 43, 15, 1}, {1, 12, 42, 84, 105, 84, 42, 12, 1}, {1, 19, 54, 126, 189, 189, 126, 54, 19, 1}, {1, 15, 91, 180, 315, 378, 315, 180, 91, 15, 1}

Mathematica

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

Crossrefs

Sequence in context: A263861 A357940 A133800 * A132733 A347971 A082039

Adjacent sequences: A146897 A146898 A146899 * A146901 A146902 A146903

Keyword

nonn

Author

Roger L. Bagula, Nov 02 2008

Status

approved