A115264 Diagonal sums of correlation triangle for floor((n+2)/2).

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

Offset

0,3

Comments

Diagonal sums of A115263.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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.

Sequence in context: A165272 A310010 A294085 * A210631 A212543 A355193

Adjacent sequences: A115261 A115262 A115263 * A115265 A115266 A115267

Keyword

easy,nonn

Author

Paul Barry, Jan 18 2006

Status

approved