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%I #7 Oct 17 2014 23:21:12
%S 1,2,1,5,4,1,16,17,7,1,65,84,45,11,1,326,485,309,100,16,1,1957,3236,
%T 2339,909,196,22,1,13700,24609,19609,8702,2281,350,29,1,109601,210572,
%U 181481,89225,26950,5081,582,37,1,986410,2004749,1843901,984506,331775
%N Triangular array of coefficients of polynomials q(n,k) defined in Comments.
%C q(n,x) = 1 + k+x + (k+x)(k-1+x) + (k+x)(k-1+x)(k-2+x) + ... + (k+x)(k-1+x)(k-2+x)...(1+x). The arrays at A248229 and A248664 have the same first column, given by A000522(n) for n >= 0. The alternating row sums of the array at A248669 are also given by A000522; viz., q(n,-1) = q(n-1,0) = A000522(n-2) for n >= 2. Column 2 of A248669 is given by A093344(n) for n >= 1.
%H Clark Kimberling, <a href="/A248669/b248669.txt">Table of n, a(n) for n = 1..5000</a>
%F q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.
%e The first six polynomials:
%e p(1,x) = 1
%e p(2,x) = 2 + x
%e p(3,x) = 5 + 4 x + x^2
%e p(4,x) = 16 + 17 x + 7 x^2 + x^3
%e p(5,x) = 65 + 8 x + 45 x^2 + 11 x^3 + x^4
%e p(6,x) = 326 + 485 x + 309 x^2 + 100 x^3 + 16 x^4 + x^5
%e First six rows of the triangle:
%e 1
%e 2 1
%e 5 4 1
%e 16 17 7 1
%e 65 84 45 11 1
%e 326 485 309 100 16 1
%t t[x_, n_, k_] := t[x, n, k] = Product[x + n - i, {i, 1, k}];
%t q[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
%t TableForm[Table[q[x, n], {n, 1, 6}]];
%t TableForm[Table[Factor[q[x, n]], {n, 1, 6}]];
%t c[n_] := c[n] = CoefficientList[q[x, n], x];
%t TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
%t Flatten[Table[c[n], {n, 1, 12}]] (* A248669 sequence *)
%Y Cf. A248665, A248666, A248667, A248668, A248670.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Oct 11 2014
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