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A159040 A triangle of polynomial coefficients: p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i*A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x).

%I A159040 #4 Jan 26 2021 03:44:54
%S A159040 1,1,1,1,-4,1,1,-6,-6,1,1,-8,11,-8,1,1,-10,19,19,-10,1,1,-12,29,-40,
%T A159040 29,-12,1,1,-14,41,-70,-70,41,-14,1,1,-16,55,-112,139,-112,55,-16,1,1,
%U A159040 -18,71,-168,251,251,-168,71,-18,1,1,-20,89,-240,419,-504,419,-240,89,-20,1
%N A159040 A triangle of polynomial coefficients: p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i*A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x).
%C A159040 Row sums are:
%C A159040 {1, 2, -2, -10, -3, 20, -4, -84, -5, 274, -6,...}.
%F A159040 p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i*A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x);
%F A159040 t(n,m)=coefficients(p(x,n),x)
%e A159040 {1},
%e A159040 {1, 1},
%e A159040 {1, -4, 1},
%e A159040 {1, -6, -6, 1},
%e A159040 {1, -8, 11, -8, 1},
%e A159040 {1, -10, 19, 19, -10, 1},
%e A159040 {1, -12, 29, -40, 29, -12, 1},
%e A159040 {1, -14, 41, -70, -70, 41, -14, 1},
%e A159040 {1, -16, 55, -112, 139, -112, 55, -16, 1},
%e A159040 {1, -18, 71, -168, 251, 251, -168, 71, -18, 1},
%e A159040 {1, -20, 89, -240, 419, -504, 419, -240, 89, -20, 1}
%t A159040 Clear[A, p, n, i];
%t A159040 A[n_, 0] := 1;
%t A159040 A[n_, n_] := 1;
%t A159040 A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + 1;
%t A159040 p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);
%t A159040 Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
%t A159040 Flatten[%]
%Y A159040 A109128
%K A159040 sign,tabl,uned
%O A159040 0,5
%A A159040 _Roger L. Bagula_, Apr 03 2009