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

1, 1, 2, 1, 4, 5, 1, 7, 17, 16, 1, 11, 45, 84, 65, 1, 16, 100, 309, 485, 326, 1, 22, 196, 909, 2339, 3236, 1957, 1, 29, 350, 2281, 8702, 19609, 24609, 13700, 1, 37, 582, 5081, 26950, 89225, 181481, 210572, 109601, 1, 46, 915, 10319, 72679, 331775, 984506

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• tabl
• easy

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). (See A248669.)

Examples

			The first six polynomials:
q(1,x) = 1
q(2,x) = x + 2
q(3,x) = x^2 + 4 x + 5
q(4,x) = x^3 + 7 x^2 + 17 x + 16
q(5,x) = x^4 + 11 x^3 + 45 x^2 + 8 x + 65
q(6,x) = x^5 + 16 x^4 + 100 x^3 + 309 x^2 + 485 x + 326
First six rows of the triangle:
1
1   2
1   4    5
1   7    17   16
1   11   45   84   65
1   16   100  309  485  326
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A248665, A248666, A248667, A248668, A248669.

Programs

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] = Reverse[CoefficientList[q[x, n], x]];
TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
Flatten[Table[c[n], {n, 1, 12}]]   (* A248669 sequence *)

Formula

q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.