A045513 -id:A045513 - cp's OEIS frontend

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

Paul Barry, Jan 18 2006

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
a(n-4) seems to be the number of face-magic cubes or order 2 with magic sum n, which means the sum of the 4 numbers at the 4 corners of each of the 6 faces equals n. (The 8 integers at the corners do not need to be distinct; copies by the 48 operations of rotations and flips are counted only once, cf. A203286, A381589. All 8 integers are positive.). E.g., 1=a(4-4) is the cube with magic sum 4, placing 1 at each corner. 1 =a(5-4) is the number of cubes with magic sum 5 obtained by placing 1 at 6 of the 8 corners but 2 at two corners opposite along a space diagonal. - R. J. Mathar, Mar 11 2025

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).

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.

R:=PowerSeriesRing(Integers(), 0); Coefficients(R!( 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Jan 13 2020

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

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 *)

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 */

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

x=PowerSeriesRing(QQ,'x').gen(); 1/((1-x)*(1-x**2)**2*(1-x**3)*(1-x**4))

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

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

Original entry on oeis.org

1, 3, 9, 20, 42, 78, 139, 231, 372, 573, 861, 1254, 1791, 2499, 3432, 4629, 6162, 8085, 10492, 13455, 17094, 21503, 26832, 33201, 40795, 49764, 60333, 72687, 87096, 103785, 123075, 145236, 170646, 199626, 232617, 269997, 312277, 359898, 413448, 473438

Offset: 1

Alford Arnold, Aug 21 2009

Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),
or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,
or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.
If we apply the enumeration of Molien series as described in A139672,
this is row 45=9*5 of a table of values related to Molien series, i.e., the
product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).
This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:
      1
      |
      2
      |
1--2--3--2--1

			To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A139672 (row 21).
For G2, the corresponding sequence is A001399.
For F4, the corresponding sequence is A115264.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210634 for a closely related sequence.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^3*(1-x^2)^3*(1-x^3)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(x/((1-x)^3*(1-x^2)^3*(1-x^3)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Jan 13 2020

Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

x=PowerSeriesRing(QQ, 'x', 40).gen()
1/((1-x)^3*(1-x^2)^3*(1-x^3))

a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - R. J. Mathar, Mar 19 2012

Edited and extended by R. J. Mathar, Aug 22 2009

Corrected link to index entries - R. J. Mathar, Aug 26 2009

A210068 Expansion of 1/((1-x)^2(1-x^2)^3(1-x^3)^2*(1-x^4)).

Original entry on oeis.org

1, 2, 6, 12, 25, 44, 79, 128, 208, 318, 483, 704, 1019, 1430, 1992, 2712, 3664, 4862, 6407, 8320, 10735, 13686, 17344, 21760, 27153, 33592, 41353, 50532, 61468, 74290, 89415, 107008, 127576, 151332, 178882, 210496, 246898, 288420, 335920

Offset: 0

F. Chapoton, Mar 17 2012

This is associated with the root system E7, and can be described using the additive function on the affine E7 diagram:
         2
         |
1--2--3--4--3--2--1

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 2, -4, -3, 0, 7, 4, -5, -4, -5, 4, 7, 0, -3, -4, 2, 2, -1).

For G2, the corresponding sequence is A001399.
For F4, the corresponding sequence is A115264.
For E6, the corresponding sequence is A164680.
For E8, the corresponding sequence is A045513.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Jan 13 2020

CoefficientList[Series[1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{2,2,-4,-3,0,7,4,-5,-4,-5,4,7,0,-3,-4,2,2,-1},{1,2,6,12,25,44,79,128,208,318,483,704,1019,1430,1992,2712,3664,4862},40] (* Harvey P. Dale, Sep 24 2021 *)

Vec(1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4))+O(x^40)) \\ Charles R Greathouse IV, Sep 26 2012

x=PowerSeriesRing(QQ,'x',40).gen()
1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4))

G.f.: 1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)).

A045514 Number of conjugacy classes of elements of order n in E_8(C).

Original entry on oeis.org

0, 1, 2, 4, 7, 14, 20, 38, 53, 85, 118, 186, 236, 363, 464, 651, 839, 1172, 1433, 1975, 2408, 3175, 3892, 5082, 6034, 7810, 9293, 11682, 13854, 17341, 20146, 25079, 29129, 35476, 41182, 49892, 57093, 68969, 78778, 93660, 106807, 126671, 142855, 168794, 190171

Offset: 0

N. J. A. Sloane

nonn

Arjeh M. Cohen and Robert L. Griess Jr., On finite simple subgroups of the complex Lie group of type E_8, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 367-405, Proc. Sympos. Pure Math., 47, Part 2, Amer. Math. Soc., Providence, RI, 1987.

Cf. A045513.

a(n) = Sum_{d|n} mu(n/d) * [x^d] b(x), n > 0, where b(x) is the g.f. for A045513. - Sean A. Irvine, Mar 12 2021

More terms from Sean A. Irvine, Mar 12 2021

cp's OEIS Frontend

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

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A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

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A210068 Expansion of 1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)).

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Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A045514 Number of conjugacy classes of elements of order n in E_8(C).

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Author

Keywords

Links

Crossrefs

Formula

Extensions

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

A210068 Expansion of 1/((1-x)^2(1-x^2)^3(1-x^3)^2*(1-x^4)).