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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. 2

%I

%S 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 909,2339,3236,1957,1,29,350,2281,8702,19609,24609,13700,1,37,582,

%U 5081,26950,89225,181481,210572,109601,1,46,915,10319,72679,331775,984506

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

%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). (See A248669.)

%H Clark Kimberling, <a href="/A248670/b248670.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 q(1,x) = 1

%e q(2,x) = x + 2

%e q(3,x) = x^2 + 4 x + 5

%e q(4,x) = x^3 + 7 x^2 + 17 x + 16

%e q(5,x) = x^4 + 11 x^3 + 45 x^2 + 8 x + 65

%e q(6,x) = x^5 + 16 x^4 + 100 x^3 + 309 x^2 + 485 x + 326

%e First six rows of the triangle:

%e 1

%e 1 2

%e 1 4 5

%e 1 7 17 16

%e 1 11 45 84 65

%e 1 16 100 309 485 326

%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] = Reverse[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, A248669.

%K nonn,tabl,easy

%O 1,3

%A _Clark Kimberling_, Oct 11 2014

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Last modified November 21 14:18 EST 2019. Contains 329371 sequences. (Running on oeis4.)