A164680 -id:A164680 - 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

A045513 Expansion of 1/((1-x)(1-x^2)^2(1-x^3)^2(1-x^4)^2(1-x^5)*(1-x^6)).

Original entry on oeis.org

1, 1, 3, 5, 10, 15, 27, 39, 63, 90, 135, 187, 270, 364, 505, 670, 902, 1173, 1545, 1976, 2550, 3218, 4081, 5083, 6357, 7825, 9659, 11772, 14366, 17342, 20956, 25080, 30031, 35667, 42357, 49945, 58881

Offset: 0

N. J. A. Sloane

This is associated with the root system E8, and can be described using the additive function on the affine E8 diagram:
      3
      |
2--4--6--5--4--3--2--1

G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
Kaiwen Sun and Haowu Wang, Weyl invariant E8 Jacobi forms and E-strings, arXiv:2109.10578 [math.NT], 2021. See Table 1 p. 9.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1,-4,-1,0,3,6,1,0,-4,-5,-5,0,5,5,4,0,-1,-6,-3,0,1,4,1,0,-2,-1,1).

For G2, the corresponding sequence is A001399.
For D4, the corresponding sequence is A001752.
For F4, the corresponding sequence is A115264.
For E6, the corresponding sequence is A164680.
For E7, the corresponding sequence is A210068.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 13 2020

CoefficientList[Series[1/((1-x)(1-x^2)^2(1-x^3)^2(1-x^4)^2(1-x^5)(1-x^6)),{x,0,40}],x] (* Harvey P. Dale, Sep 16 2019 *)

Vec(1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).

A165188 Interleaving of A014125 and zero followed by A014125.

Original entry on oeis.org

1, 0, 3, 1, 6, 3, 11, 6, 18, 11, 27, 18, 39, 27, 54, 39, 72, 54, 94, 72, 120, 94, 150, 120, 185, 150, 225, 185, 270, 225, 321, 270, 378, 321, 441, 378, 511, 441, 588, 511, 672, 588, 764, 672, 864, 764, 972, 864, 1089, 972, 1215, 1089, 1350, 1215, 1495, 1350

Offset: 1

Alford Arnold, Sep 13 2009

nonn

This sequence convolved with A000217 (without initial term 0) yields A164680.
See A164680 for similar convolutions.
A165188 convolved with A000217 yields sequence A164680. This is to be expected since A000217 can be associated with partition 1+1+1, A164680 with partition 1+1+1+2+2+2+3 and A165188 with partition 2+2+2+3 by observing their unreduced generating functions and verified by generating the sequences by converting the partitions into finite sequences and using Euler's Transform. Thus partition 1+1+1 yields the finite sequence (3); partition 2+2+2+3 yields the finite sequence (0,3,1); and, when combined, partition 1+1+1+2+2+2+3 yields (3,3,1). - Alford Arnold, Sep 24 2009

			A014125 begins 1,3,6,11,18,27,..., thus this sequence begins 1,0,3,1,6,3,11,6,18,11,27,18,... .
G.f. = x + 3*x^3 + x^4 + 6*x^5 + 3*x^6 + 11*x^7 + 6*x^8 + 18*x^9 + 11*x^10 + ...

Index entries for linear recurrences with constant coefficients, signature (0,3,1,-3,-3,1,3,0,-1).

Cf. A014125, A000217, A164680, A001400.

I:=[1,0,3,1,6,3,11,6]; [n le 8 select I[n] else -Self(n-1)+2*Self(n-2)+3*Self(n-3)-3*Self(n-5)-2*Self(n-6)+Self(n-7)+Self(n-8)+1: n in [1..60]]; // Vincenzo Librandi, Jun 24 2015

a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -7 - n]; s SeriesCoefficient[ x / ((1 - x^2)^3 (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Feb 01 2015 *)
LinearRecurrence[{0,3,1,-3,-3,1,3,0,-1},{1,0,3,1,6,3,11,6,18},60] (* Harvey P. Dale, Apr 14 2018 *)

/* first computes u = A014125 as second bisection of A001400, then interleaves */ {m=28; u=vector(m, n, polcoeff(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))+O(x^(2*n)), 2*n-1)); vector(2*m, k, if(k%2==1,u[(k+1)/2], if(k==2, 0, u[k/2-1])))} /* Klaus Brockhaus, Sep 15 2009 */

{a(n) = my(s=1); if( n<0, s=-1; n=-7-n); s * polcoeff( x / ((1 - x^2)^3 * (1 - x^3)) + x * O(x^n), n)}; /* Michael Somos, Feb 01 2015 */

a(n) = -a(n-1)+2*a(n-2)+3*a(n-3)-3*a(n-5)-2*a(n-6)+a(n-7)+a(n-8)+1 for n > 8; a(1)=1, a(2)=0, a(3)=3, a(4)=1, a(5)=6, a(6)=3, a(7)=11, a(8)=6. - Klaus Brockhaus, Sep 15 2009
G.f.: x/((1-x)^4*(1+x)^3*(1+x+x^2)). - Klaus Brockhaus, Sep 15 2009
a(n) = (2*n^3+21*n^2+63*n+49)/288-(-1)^n*(9+7*n+n^2)/32+A057078(n)/9. - R. J. Mathar, Sep 17 2009
Euler transform of length 3 sequence [ 0, 3, 1]. - Michael Somos, Feb 01 2015
G.f.: x / ((1 - x^2)^3 * (1 - x^3)). - Michael Somos, Feb 01 2015
a(n) = -a(-7 - n) for all n in Z.
a(n+3) - a(n) = 0 if n odd else (n+6) * (n+4) / 8. - Michael Somos, Feb 01 2015
a(2*n + 1) = a(2*n + 4) = A014125(n) for all n in Z. - Michael Somos, Feb 01 2015

Edited and extended by Klaus Brockhaus, Sep 15 2009

Linear recurrence link and signature correct by Michel Marcus, Jun 25 2015

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)).

A210634 G.f. for Ehrhart quasi-polynomials for hyperplane arrangements of type E_6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 540540, 615342, 698608

Offset: 0

N. J. A. Sloane, Mar 25 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Andreas Blass, Bruce E. Sagan, Characteristic and Ehrhart polynomials, arXiv:math/9801008 [math.CO], 1998.
Andreas Blass, Bruce E. Sagan, Characteristic and Ehrhart polynomials, J. Algebraic Combin. 7 (1998), no. 2, 115--126. MR1609889 (99c:05204)
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

A164680 is similar but has a different offset.

LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{0,0,0,0,0,0,0,0,0,0,0,0,1},60] (* Harvey P. Dale, Mar 27 2025 *)

G.f.: x^12*f(1)^3*f(2)^3*f(3) where f(k)=1/(1-x^k).

G.f.: x^12/((1-x)^3*(1-x^2)^3*(1-x^3)). - Colin Barker, Jul 22 2013

A164678 Convolve A008619 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

Original entry on oeis.org

1, 2, 1, 5, 4, 2, 9, 8, 6, 2, 17, 16, 14, 9, 3, 28, 27, 25, 20, 12, 3, 47, 46, 44, 39, 30, 16, 4, 73, 72, 70, 65, 56, 40, 20, 4, 114, 113, 111, 106, 97, 80, 55, 25, 5, 170, 169, 167, 162, 153, 136, 109, 70, 30, 5, 253, 252, 250, 245, 236, 219, 191, 147, 91, 36, 6, 365

Offset: 1

Alford Arnold, Aug 25 2009

Sequence A164680 is not in the triangle since 45 = 3*15 and 15 is not a member of A000040 (the prime numbers).

			The desired triangle begins:
.1
.2..1
.5..4..2
.9..8..6..2
17.16.14..9..3
28.27.25.20.12..3
47.46.44.39.30.16..4
etc.
Note that 21 = 3*7 maps to 1,2,5,9,17,27,44,... A139672 is embedded in the triangle.

Cf. A008619 A002620 A006918 A097701 A139672 ... A000097 (embedded sequences).

cp's OEIS Frontend

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

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A045513 Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).

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A165188 Interleaving of A014125 and zero followed by A014125.

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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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A210634 G.f. for Ehrhart quasi-polynomials for hyperplane arrangements of type E_6.

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A164678 Convolve A008619 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

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A045513 Expansion of 1/((1-x)(1-x^2)^2(1-x^3)^2(1-x^4)^2(1-x^5)*(1-x^6)).

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