|
|
A115264
|
|
Diagonal sums of correlation triangle for floor((n+2)/2).
|
|
7
|
|
|
1, 1, 3, 4, 8, 10, 17, 21, 32, 39, 55, 66, 89, 105, 136, 159, 200, 231, 284, 325, 392, 445, 528, 595, 697, 780, 903, 1005, 1152, 1275, 1449, 1596, 1800, 1974, 2211, 2415, 2689, 2926, 3240, 3514, 3872, 4186, 4592, 4950, 5408, 5814, 6328, 6786, 7361
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
This is associated with the root system F4, and can be described using the additive function on the affine F4 diagram:
2--4--3--2--1
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1,-2,0,2,1,1,-2,-1,1).
|
|
FORMULA
|
G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} [j<=k]*floor((k-j+2)/2)*[j<=n-2k]*floor((n-2k-j+2)/2).
a(n) = A099837(n+3)/27 + A056594(n)/16 + (-1)^n*(2*n^2 +24*n +63)/256 +(6*n^4 +144*n^3 +1194*n^2 +3960*n +4267)/6912 . - R. J. Mathar, Mar 19 2012
|
|
MAPLE
|
seq(coeff(series(1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Jan 13 2020
|
|
MATHEMATICA
|
CoefficientList[Series[1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)), {x, 0, 50}], x] (* G. C. Greubel, Jan 13 2020 *)
|
|
PROG
|
(Sage) x=PowerSeriesRing(QQ, 'x').gen(); 1/((1-x)*(1-x**2)**2*(1-x**3)*(1-x**4))
(Maxima) A115264(n) := block( A099837(n+3)/27 + A056594(n)/16+(-1)^n*(2*n^2+24*n+63)/256 +(6*n^4 +144*n^3+1194*n^2+3960*n+4267)/6912 )$ /* R. J. Mathar, Mar 19 2012 */
(PARI) my(x='x+O('x^50)); Vec(1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4))) \\ G. C. Greubel, Jan 13 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 0); Coefficients(R!( 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Jan 13 2020
|
|
CROSSREFS
|
For G2, the corresponding sequence is A001399.
For E6, the corresponding sequence is A164680.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210631 for a very similar sequence.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|