|
|
A156918
|
|
Triangle formed by coefficients of the expansion of p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (2*j+1)^n*(-x + x^2)^j.
|
|
5
|
|
|
1, 1, -1, 1, 1, -6, 7, -2, 1, 1, -23, 46, -47, 26, -3, 1, 1, -76, 306, -536, 459, -232, 82, -4, 1, 1, -237, 1919, -5046, 6965, -5995, 3109, -958, 247, -5, 1, 1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1, 1, -2179, 62836, -381037, 1099549, -1878718, 2123525, -1658537, 898985, -346886, 93377, -13109, 2200, -7, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
Row sums are one.
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n + 1)*Sum_{j >= 0} (2*j+1)^n*(-x + x^2)^j.
|
|
EXAMPLE
|
Irregular triangle begins as:
1;
1, -1, 1;
1, -6, 7, -2, 1;
1, -23, 46, -47, 26, -3, 1;
1, -76, 306, -536, 459, -232, 82, -4, 1;
1, -237, 1919, -5046, 6965, -5995, 3109, -958, 247, -5, 1;
1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1;
|
|
MATHEMATICA
|
p[x_, n_] = (1+x-x^2)^(n+1)*Sum[(2*k+1)^n*(-x+x^2)^k, {k, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
|
|
PROG
|
(Sage)
def T(n, k): return ( (1+x-x^2)^(n+1)*sum((2*j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
flatten([1]+[[T(n, k) for k in (0..2*n)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,tabf
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|