A115264 -id:A115264 - cp's OEIS frontend

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

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

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)).

A203286 Number of arrays of 2n nondecreasing integers in -3..3 with sum zero and equal numbers greater than zero and less than zero.

Original entry on oeis.org

4, 12, 28, 57, 104, 176, 280, 425, 620, 876, 1204, 1617, 2128, 2752, 3504, 4401, 5460, 6700, 8140, 9801, 11704, 13872, 16328, 19097, 22204, 25676, 29540, 33825, 38560, 43776, 49504, 55777, 62628, 70092, 78204, 87001, 96520, 106800, 117880, 129801

Offset: 1

R. H. Hardin, Dec 31 2011

nonn

Column 3 of A203291.

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 separately. All 8 integers are positive.). E.g., 4 =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 to each other along a space diagonal (with 4 different space diagonals available). See also A381589 and A115264. - R. J. Mathar, Mar 11 2025

			Some solutions for n=3:
.-2...-2...-2...-2...-3...-3...-3...-3...-1...-3....0...-2...-1...-3...-2...-3
..0...-2...-2...-1....0...-3...-1...-1...-1...-2....0...-2...-1...-1...-2...-2
..0...-2....0...-1....0...-2....0...-1...-1...-1....0....0....0...-1...-1...-2
..0....1....0....1....0....2....0....1....1....1....0....0....0....1....1....2
..0....2....1....1....0....3....2....2....1....2....0....2....1....1....2....2
..2....3....3....2....3....3....2....2....1....3....0....2....1....3....2....3

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A203291, A005994.

Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6).
Conjectures from Colin Barker, Jun 04 2018: (Start)
G.f.: x*(4 - 4*x + 5*x^3 - 4*x^4 + x^5) / ((1 - x)^5*(1 + x)).
a(n) = (48 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n even.
a(n) = (42 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n odd.
(End)

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)).

A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 0, 2, 1, 4, 2, 7, 4, 11, 7, 16, 11, 23, 16, 31, 23, 41, 31, 53, 41, 67, 53, 83, 67, 102, 83, 123, 102, 147, 123, 174, 147, 204, 174, 237, 204, 274, 237, 314, 274, 358, 314, 406, 358, 458, 406, 514, 458, 575, 514, 640, 575, 710, 640, 785, 710, 865, 785, 950

Offset: 0

Michael Somos, Feb 02 2015

Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2.
The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, u+v >= x+w, and x+u+v+w is even.
Euler transform of length 4 sequence [ 0, 2, 1, 1].

			G.f. = 1 + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + 7*x^6 + 4*x^7 + 11*x^8 + 7*x^9 + ...

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

Cf. A001400, A002620, A005044, A115264, A165188.

Cf. A241526, A000601, A181120.

I:=[1,0,2,1,4,2,7,4,11,7,16]; [n le 11 select I[n] else 2*Self(n-2)+Self(n-3)-2*Self(n-5)-2*Self(n-6)+Self(n-8)+2*Self(n-9)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Feb 03 2015

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288];
a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -11 - n]; s SeriesCoefficient[ 1 / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, u + v >= x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[1 / (1 - 2 x^2 - x^3 + 2 x^5 + 2 x^6 - x^8 - 2 x^9 + x^11), {x, 0, 60}], x] (* Vincenzo Librandi, Feb 03 2015 *)

{a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288};

{a(n) = my(s=1); if( n<0, s=-1; n=-11-n); s * polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
a(n) = -a(-11-n) for all n in Z.
a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16).
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(n) - a(n-2) = A005044(n+3) for all n in Z.
a(n) + a(n-1) = A001400(n) for all n in Z.
a(n) + a(n-2) = A165188(n+1) for all n in Z.
a(n) = A115264(n) - A115264(n-1) for all n in Z.
a(2*n) - a(2*n-6) = a(2*n+3) - a(2*n-3) = A002620(n+2) for all n in Z. - Michael Somos, Feb 11 2015
a(n) = (2*n^3+33*n^2+181*n+234+3*(3*n^2+33*n+86)*(-1)^n+84*(-1)^((2*n+1-(-1)^n)/4)-96*((1+(-1)^n)*floor(((2*n+9+(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24))+(1-(-1)^n)*floor(((2*n+5+(-1)^n-6*(-1)^((2*n-1+(-1)^n)/4))/24))))/576. - Luce ETIENNE, May 22 2015

A381589 The number of face-magic cubes with magic sum n and distinct positive integers at the vertices including 1.

Original entry on oeis.org

3, 2, 6, 6, 16, 13, 21, 28, 38, 40, 57, 58, 81, 92, 108, 118, 150, 158, 188, 213, 242, 257, 309, 324, 373, 408, 448, 483, 551, 578, 643, 695, 759, 804, 894, 935, 1023, 1097, 1177, 1243, 1360, 1416, 1528, 1625, 1731, 1816, 1959, 2041, 2181, 2300, 2430, 2541, 2721, 2822, 2992, 3141, 3300, 3441, 3650, 3781, 3985, 4163, 4358, 4526, 4777, 4934

Offset: 18

R. J. Mathar, Mar 12 2025

nonn

The face-magic cubes counted here have 8 distinct positive integers (including 1) at the vertices, and each sum over the 4 vertices of the 6 faces is the same. Cubes obtained by rotations or mirrors of the octahedral point group are counted only once.

			The 3 face-magic cubes with sum 18 are 1 4 5 8 - 6 7 2 3, 1 4 5 8 - 7 6 3 2 and 1 6 3 8 - 7 4 5 2, values at the base and values at the top face separated by a dash.

R. J. Mathar, Illustrations, conjectured g.f.

Cf. A203286, A115264.

A115263 Correlation triangle for floor((n+2)/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 3, 4, 6, 4, 3, 3, 5, 7, 7, 5, 3, 4, 6, 10, 10, 10, 6, 4, 4, 7, 11, 13, 13, 11, 7, 4, 5, 8, 14, 16, 19, 16, 14, 8, 5, 5, 9, 15, 19, 22, 22, 19, 15, 9, 5, 6, 10, 18, 22, 28, 28, 28, 22, 18, 10, 6

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A096338. Diagonal sums are A115264. T(2n,n) is A005993. T(2n,n)-T(2n,n+1) is floor((n+2)/2)(1+(-1)^n)/2 (aerated n+1).

			Triangle begins
1;
1,1;
2,2,2;
2,3,3,2;
3,4,6,4,3;
3,5,7,7,5,3;

G.f.: (1+x)(1+xy)/((1-x^2)^2*(1-x^2*y^2)^2*(1-x^2*y)); Number triangle T(n, k)=sum{j=0..n, [j<=k]*floor((k-j+2)/2)*[j<=n-k]*floor((n-k-j+2)/2)}.

A210631 G.f. for Ehrhart quasi-polynomials for hyperplane arrangements of type F_4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 7875, 8515, 9090, 9800, 10440

Offset: 0

N. J. A. Sloane, Mar 25 2012

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 (1,2,-1,-1,-2,0,2,1,1,-2,-1,1).

Similar to A115264 but has different offset.

LinearRecurrence[{1,2,-1,-1,-2,0,2,1,1,-2,-1,1},{0,0,0,0,0,0,0,0,0,0,0,0,1},70] (* Harvey P. Dale, Oct 28 2015 *)

G.f.: x^12*f(1)*f(2)^2*f(3)*f(4) where f(k)=1/(1-x^k).

G.f.: x^12/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)). - Colin Barker, Jul 22 2013

cp's OEIS Frontend

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

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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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A203286 Number of arrays of 2n nondecreasing integers in -3..3 with sum zero and equal numbers greater than zero and less than zero.

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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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A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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A381589 The number of face-magic cubes with magic sum n and distinct positive integers at the vertices including 1.

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A115263 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)).

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)).