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 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}]]. 0

%I

%S 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,

%T 1,23,65,107,107,65,23,1,1,16,56,112,140,112,56,16,1,1,29,72,168,252,

%U 252,168,72,29,1,1,20,137,240,420,504,420,240,137,20,1

%N 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}]].

%C 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.

%F 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)).

%e {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}

%t 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[%]

%K nonn

%O 0,5

%A _Roger L. Bagula_, Nov 02 2008

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Last modified December 2 11:44 EST 2021. Contains 349440 sequences. (Running on oeis4.)