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%I #20 Sep 08 2022 08:45:38
%S 1,1,1,1,3,1,1,6,6,1,1,12,18,12,1,1,25,50,50,25,1,1,54,135,180,135,54,
%T 1,1,119,357,595,595,357,119,1,1,264,924,1848,2310,1848,924,264,1,1,
%U 585,2340,5460,8190,8190,5460,2340,585,1,1,1290,5805,15480,27090,32508
%N Coefficients of polynomial P(n) by rows, with P(n) = (x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1) for n > 0 and P(0) = 1.
%C Original name: A new symmetrical polynomial form to give a triangle sequence: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
%C Row sums are:{1, 2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840}.
%C Row sums are 1 and (6*2^k + 4^k)/8 for k >= 1 (see A257273). - _Robert Israel_, Apr 29 2015
%F G.f.: y/(4*(2*y-1)) - 1/(x*y+y-1) - 1/(8*(2*x*y+2*y-1)) + 1/(8*(2*x*y-1)). - _Robert Israel_, Apr 29 2015
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 6, 1;
%e 1, 12, 18, 12, 1;
%e 1, 25, 50, 50, 25, 1;
%e 1, 54, 135, 180, 135, 54, 1;
%e 1, 119, 357, 595, 595, 357, 119, 1;
%e 1, 264, 924, 1848, 2310, 1848, 924, 264, 1;
%e 1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1;
%e 1, 1290, 5805, 15480, 27090, 32508, 27090, 15480, 5805, 1290, 1;
%e ...
%t p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[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[%]
%o (Magma) /* As triangle: */ [1]; for n in [1..10] do; R<x> := PolynomialAlgebra(RationalField(), n); Coefficients((x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1)); end for; // _Bruno Berselli_, Apr 30 2015
%Y Cf. A257273 (row sums).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 02 2008
%E New name from _Charles R Greathouse IV_, Apr 29 2015