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

1, 2, 1, 5, 4, 1, 16, 17, 7, 1, 65, 84, 45, 11, 1, 326, 485, 309, 100, 16, 1, 1957, 3236, 2339, 909, 196, 22, 1, 13700, 24609, 19609, 8702, 2281, 350, 29, 1, 109601, 210572, 181481, 89225, 26950, 5081, 582, 37, 1, 986410, 2004749, 1843901, 984506, 331775

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

Cf. A248665, A248666, A248667, A248668, A248670.

Sequence in context: A171515 A110271 A073107 * A103718 A113350 A227372

Adjacent sequences: A248666 A248667 A248668 * A248670 A248671 A248672

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 11 2014

Status

approved