cp's OEIS Frontend

A248669 Triangular array of coefficients of polynomials q(n,k) defined in Comments.

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