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

1,3

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.)

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:
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

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 *)

Crossrefs

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

Sequence in context: A121289 A211561 A134248 * A080935 A362926 A102661

Adjacent sequences: A248667 A248668 A248669 * A248671 A248672 A248673

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 11 2014

Status

approved