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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
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 (list; table; graph; refs; listen; history; text; internal format)
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 A102661 A121574

Adjacent sequences:  A248667 A248668 A248669 * A248671 A248672 A248673

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Oct 11 2014

STATUS

approved

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Last modified May 29 21:22 EDT 2017. Contains 287257 sequences.