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A248670 Triangular array of coefficients of polynomials q defined in Comments; the coefficients are written in the order of decreasing powers of x.

%I A248670 #7 Oct 18 2014 00:23:02
%S A248670 1,1,2,1,4,5,1,7,17,16,1,11,45,84,65,1,16,100,309,485,326,1,22,196,
%T A248670 909,2339,3236,1957,1,29,350,2281,8702,19609,24609,13700,1,37,582,
%U A248670 5081,26950,89225,181481,210572,109601,1,46,915,10319,72679,331775,984506
%N A248670 Triangular array of coefficients of polynomials q defined in Comments; the coefficients are written in the order of decreasing powers of x.
%C A248670 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).  (See A248669.)
%H A248670 Clark Kimberling, <a href="/A248670/b248670.txt">Table of n, a(n) for n = 1..5000</a>
%F A248670 q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.
%e A248670 The first six polynomials:
%e A248670 q(1,x) = 1
%e A248670 q(2,x) = x + 2
%e A248670 q(3,x) = x^2 + 4 x + 5
%e A248670 q(4,x) = x^3 + 7 x^2 + 17 x + 16
%e A248670 q(5,x) = x^4 + 11 x^3 + 45 x^2 + 8 x + 65
%e A248670 q(6,x) = x^5 + 16 x^4 + 100 x^3 + 309 x^2 + 485 x + 326
%e A248670 First six rows of the triangle:
%e A248670 1
%e A248670 1   2
%e A248670 1   4    5
%e A248670 1   7    17   16
%e A248670 1   11   45   84   65
%e A248670 1   16   100  309  485  326
%t A248670 t[x_, n_, k_] := t[x, n, k] = Product[x + n - i, {i, 1, k}];
%t A248670 q[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
%t A248670 TableForm[Table[q[x, n], {n, 1, 6}]];
%t A248670 TableForm[Table[Factor[q[x, n]], {n, 1, 6}]];
%t A248670 c[n_] := c[n] = Reverse[CoefficientList[q[x, n], x]];
%t A248670 TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
%t A248670 Flatten[Table[c[n], {n, 1, 12}]]   (* A248669 sequence *)
%Y A248670 Cf. A248665, A248666, A248667, A248668, A248669.
%K A248670 nonn,tabl,easy
%O A248670 1,3
%A A248670 _Clark Kimberling_, Oct 11 2014