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

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). 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.

Examples

			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
		

Links

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

Crossrefs

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

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] = 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.