A146898 Lower polynomial approximation of Eulerian numbers: 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}]].

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 8, 12, 8, 1, 1, 17, 20, 20, 17, 1, 1, 12, 47, 40, 47, 12, 1, 1, 23, 65, 107, 107, 65, 23, 1, 1, 16, 56, 112, 140, 112, 56, 16, 1, 1, 29, 72, 168, 252, 252, 168, 72, 29, 1, 1, 20, 137, 240, 420, 504, 420, 240, 137, 20, 1

Offset

0,5

Comments

Row sums are:{1, 2, 6, 24, 30, 76, 160, 392, 510, 1044, 2140}. The effort here was to match the modulo two behavior to the Sierpinski gasket while adding a term polynomial.

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}]]; t(n,m)=coefficients(p(x,n)).

Example

{1}, {1, 1}, {1, 4, 1}, {1, 11, 11, 1}, {1, 8, 12, 8, 1}, {1, 17, 20, 20, 17, 1}, {1, 12, 47, 40, 47, 12, 1}, {1, 23, 65, 107, 107, 65, 23, 1}, {1, 16, 56, 112, 140, 112, 56, 16, 1}, {1, 29, 72, 168, 252, 252, 168, 72, 29, 1}, {1, 20, 137, 240, 420, 504, 420, 240, 137, 20, 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}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

Crossrefs

Sequence in context: A112500 A152938 A154096 * A152970 A154986 A154983

Adjacent sequences: A146895 A146896 A146897 * A146899 A146900 A146901

Keyword

nonn

Author

Roger L. Bagula, Nov 02 2008

Status

approved