OFFSET
1,2
COMMENTS
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.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..5000
FORMULA
q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.
EXAMPLE
The first six polynomials:
p(1,x) = 1
p(2,x) = 2 + x
p(3,x) = 5 + 4 x + x^2
p(4,x) = 16 + 17 x + 7 x^2 + x^3
p(5,x) = 65 + 8 x + 45 x^2 + 11 x^3 + x^4
p(6,x) = 326 + 485 x + 309 x^2 + 100 x^3 + 16 x^4 + x^5
First six rows of the triangle:
1
2 1
5 4 1
16 17 7 1
65 84 45 11 1
326 485 309 100 16 1
MATHEMATICA
t[x_, n_, k_] := t[x, n, k] = Product[x + n - i, {i, 1, k}];
q[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[q[x, n], {n, 1, 6}]];
TableForm[Table[Factor[q[x, n]], {n, 1, 6}]];
c[n_] := c[n] = CoefficientList[q[x, n], x];
TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
Flatten[Table[c[n], {n, 1, 12}]] (* A248669 sequence *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Oct 11 2014
STATUS
approved