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Symmetrical product form polynomial as triangle sequence coefficients: q(x,n)=Product[x + 2*n - i + 1, {i, 0, n - 1}]; p(x,n)=q(x,n)+x^2+x^n*q(1/x,m);t(n,m)=coefficients(p(x,n).
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%I #2 Mar 30 2012 17:34:33

%S 2,4,4,21,18,21,211,125,125,211,3025,1680,670,1680,3025,55441,31639,

%T 7960,7960,31639,55441,1235521,725655,178199,45570,178199,725655,

%U 1235521,32432401,19471584,4988326,765289,765289,4988326,19471584

%N Symmetrical product form polynomial as triangle sequence coefficients: q(x,n)=Product[x + 2*n - i + 1, {i, 0, n - 1}]; p(x,n)=q(x,n)+x^2+x^n*q(1/x,m);t(n,m)=coefficients(p(x,n).

%C Row sums are:

%C {2, 8, 60, 672, 10080, 190080, 4324320, 115315200, 3528645120, 121898649600,

%C 4693098009600,...}.

%F q(x,n)=Product[x + 2*n - i + 1, {i, 0, n - 1}];

%F p(x,n)=q(x,n)+x^2+x^n*q(1/x,m);

%F t(n,m)=coefficients(p(x,n).

%e {2},

%e {4, 4},

%e {21, 18, 21},

%e {211, 125, 125, 211},

%e {3025, 1680, 670, 1680, 3025},

%e {55441, 31639, 7960, 7960, 31639, 55441},

%e {1235521, 725655, 178199, 45570, 178199, 725655, 1235521},

%e {32432401, 19471584, 4988326, 765289, 765289, 4988326, 19471584, 32432401},

%e {980179201, 598482108, 159173510, 24219972, 4535538, 24219972, 159173510, 598482108, 980179201}, {33522128641, 20742534711, 5681716170, 904550030, 98395248, 98395248, 904550030, 5681716170, 20742534711, 33522128641},

%e {1279935820801, 800575997925, 224518871346, 37179349350, 4041096653, 595737450, 4041096653, 37179349350, 224518871346, 800575997925, 1279935820801}

%t Clear[p, a, b, c, d, n, q, q2, x];

%t q[x_, n_] = Product[x + 2*n - i + 1, {i, 0, n - 1}];

%t p[x_, n_] = q[x, n] + x^n*q[1/x, n];

%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

%t Flatten[%]

%K nonn,tabl,uned

%O 0,1

%A _Roger L. Bagula_, Jan 25 2009