login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A248669 Triangular array of coefficients of polynomials q(n,k) defined in Comments. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 26 00:44 EDT 2017. Contains 289798 sequences.