|
|
A248830
|
|
Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-2k)^k for 0 <= k <= n.
|
|
2
|
|
|
1, 3, 1, 3, 9, 1, 3, 45, 19, 1, 3, 173, 211, 33, 1, 3, 573, 1811, 633, 51, 1, 3, 1725, 13331, 9273, 1491, 73, 1, 3, 4861, 88595, 115113, 32851, 3013, 99, 1, 3, 13053, 547347, 1276329, 606291, 92613, 5475, 129, 1, 3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1, 3, 84989, 17947155, 125008041, 147521619, 52760613, 7480803, 479601, 14563, 201, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-2)^1 + A_2*(x-4)^2 + ... + A_n*(x-2n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
|
|
LINKS
|
|
|
FORMULA
|
T(n,n-1) = 2*n^2 + 1 for n > 0.
T(n,1) = 2^n*(n^2-2*n+3)-3 for n > 0.
|
|
EXAMPLE
|
1;
3, 1;
3, 9, 1;
3, 45, 19, 1;
3, 173, 211, 33, 1;
3, 573, 1811, 633, 51, 1;
3, 1725, 13331, 9273, 1491, 73, 1;
3, 4861, 88595, 115113, 32851, 3013, 99, 1;
3, 13053, 547347, 1276329, 606291, 92613, 5475, 129, 1;
3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1;
|
|
PROG
|
(PARI) for(n=0, 10, for(k=0, n, if(!k, if(n, print1(3, ", ")); if(!n, print1(1, ", "))); if(k, print1(sum(i=1, n, ((2*k)^(i-k)*i*binomial(i, k)))/k, ", "))))
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|