A007980 -id:A007980 - cp's OEIS frontend

A260253 Number of symmetry-allowed, linearly-independent terms at n-th order in the expansion of E x (e+a) rovibrational perturbation matrix H(Jx,Jy,Jz).

1, 0, 4, 1, 9, 2, 16, 4, 25, 7, 36, 10, 49, 14, 64, 19, 81, 24, 100, 30, 121, 37, 144, 44, 169, 52, 196, 61, 225, 70, 256, 80, 289, 91, 324, 102, 361, 114, 400, 127, 441, 140, 484, 154, 529, 169, 576, 184, 625, 200, 676, 217, 729, 234, 784, 252, 841, 271

Offset: 0

Bradley Klee, Jul 20 2015

a(n) are also coefficients in a Molien Series for G = H x T x D3, where H is Hermitian conjugacy, T is Time-reversal, and D3 is triangular Dihedral. |G| = 24.

Harter et al. give only one second-order term, while Sadovskií et al. give only two (see links).

W. G. Harter, H. W. Galbraith, and C. W. Patterson, Energy level cluster analysis for E(v2) vibration rotation spectrum of spherical top molecules, J. Chem. Phys, 69, 4888 (1978).
D. A. Sadovskií and B. I. Zhilinskií, Qualitative analysis of vibration-rotation Hamiltonians for spherical top molecules, Molecular Physics 65, 1 (1988).
N. J. A. Sloane, Error-correcting codes and invariant theory: new applications of a nineteenth-century technique, American Mathematical Monthly (1977): 82-107.
Richard P. Stanley, Invariants of finite groups and their applications to combinatorics, Bulletin of the American Mathematical Society 1.3 (1979): 475-511.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,1,0,-2,0,1).

Cf. A007980 (bisection), A002264, A260220, A000290 (bisection).

D[(1 + 2 x^2 + x^3 + 2 x^4 + x^6 + x^7)/((1 - x^2)^3*(1 + x^2 + x^4)), {x, #}]/#!/.x -> 0 & /@ Range[0, 30]
CoefficientList[Series[(1 + 2 x^2 + x^3 + 2 x^4 + x^6 + x^7)/((1 - x^2)^3 (1 + x^2 + x^4)), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 22 2015 *)

Vec((1 + 2 * x^2 + x^3 + 2 * x^4 + x^6 + x^7)/((1 - x^2)^3 *(1 + x^2 + x^4)) + O(x^90)) \\ Michel Marcus, Aug 05 2015

G.f.: (1 + 2 * x^2 + x^3 + 2 * x^4 + x^6 + x^7)/((1 - x^2)^3 *(1 + x^2 + x^4)).

a(n)= 2*a(n-2) -a(n-4) +a(n-6) -2*a(n-8) +a(n-10). - R. J. Mathar, Jul 20 2023

A347026 Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 17, 15, 13, 11, 9, 7, 5, 3, 1

Offset: 1

Eddie Gutierrez, Aug 11 2021

The terms of this sequence are the numbers in an irregular triangle corresponding to the addition of rows when multiplying two large numbers via a novel method (see Links).
Sums of the rising diagonals yield sequence A007980.
When the 2n terms in row n are used as the coefficients of a (2n-1)st-order polynomial in x, dividing that polynomial by x+1 produces a (2n-2)nd-order polynomial whose coefficients are the n-th row of A004737 (if that sequence is taken as an irregular triangle with 2n-1 terms in its n-th row). E.g., for n=3, (x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1)/(x+1) = x^4 + 2x^3 + 3x^2 + 2x + 1.

			Triangle begins:
  1,  1;
  1,  3,  3,  1;
  1,  3,  5,  5,  3,  1;
  1,  3,  5,  7,  7,  5,  3,  1;
  1,  3,  5,  7,  9,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 11,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 13, 13, 11,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 13, 15, 15, 13, 11,  9,  7,  5,  3,  1;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eddie Gutierrez, A Novel Method for Multiplying Large Numbers.
Eddie Gutierrez, The Column Addition Triangle (CAT)-A Pascal Analog (Part I).
Eddie Gutierrez, The Column Addition Triangle (CAT) and Polynomials.

Even-indexed rows of A157454.
Antidiagonal sums give A007980.
Row lengths give nonzero terms of A005843.
Cf. A004737.

#include 
int main()
{
   int n, k;
   for (n=1; n<=13; n++)
   {
      for (k=1; k<=n; k++)
      {
         printf("%d ", 2*k - 1);
      }
      for (k=n+1; k<=2*n; k++)
      {
         printf("%d ", 4*n - 2*k + 1);
      }
      printf("\n");
   }
   return 0;
}

Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* Michael De Vlieger, Aug 18 2021 *)
Flatten[Table[Join[Range[1,2n+1,2],Range[2n+1,1,-2]],{n,0,10}]] (* Harvey P. Dale, Aug 31 2024 *)

row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ Michel Marcus, Aug 17 2021

T(n,k) = 2k - 1 for 1 <= k <= n,

4n - 2k + 1 for n+1 <= k <= 2n.

Better definition from Omar E. Pol, Aug 14 2021

A115281 Correlation triangle for the sequence 2-0^n.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 2, 6, 6, 2, 2, 6, 9, 6, 2, 2, 6, 10, 10, 6, 2, 2, 6, 10, 13, 10, 6, 2, 2, 6, 10, 14, 14, 10, 6, 2, 2, 6, 10, 14, 17, 14, 10, 6, 2, 2, 6, 10, 14, 18, 18, 14, 10, 6, 2, 2, 6, 10, 14, 18, 21, 18, 14, 10, 6, 2

Offset: 0

Paul Barry, Jan 19 2006

Row sums are (n+1)^2 (A000290(n+1)). Diagonal sums are the Molien series A007980. T(2n,n) is 4n+1 (A016813), the partial sums of (2-0^n)^2. T(2,n)-T(2n,n+1) is 3-2*0^n.
From Mats Granvik, Jul 06 2010: (Start)
If seen as a square array:
1, 2, 2, 2
2, 5, 6, 6
2, 6, 9, 10
2, 6, 10, 13
then the matrix inverse contains the same values, only signed and in reversed order:
13, -10, 6, -2
-10, 9, -6, 2
6, -6, 5, -2
-2, 2, -2, 1
(End)

			Triangle begins
1;
2,2;
2,5,2;
2,6,6,2;
2,6,9,6,2;
2,6,10,10,6,2;

Flatten[Table[Table[If[n - k + 1 == k, 4*(n - k + 1 - 1) + 1, If[n - k + 1 > k, 4*(k - 1) + 2, 4*(n - k + 1 - 1) + 2]], {k, 1, n}], {n, 1, 11}]] (* Mats Granvik, Jan 06 2016 *)

G.f.: (1+x)(1+x*y)/((1-x)(1-x*y)(1-x^2*y)).

T(n, k) = sum{j=0..n, [j<=k]*(2-0^(k-j))*[j<=n-k]*(2-0^(n-k-j))}.

a(65)-a(66) from Mats Granvik, Jan 06 2016

A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 2, 4, 2, 0, 2, 6, 4, 2, 0, 2, 8, 8, 4, 2, 0, 2, 10, 14, 8, 4, 2, 0, 2, 12, 20, 16, 8, 4, 2, 0, 2, 14, 28, 26, 16, 8, 4, 2, 0, 2, 16, 38, 40, 28, 16, 8, 4, 2, 0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2, 0, 2, 20, 60, 84, 72, 48, 28, 16, 8, 4, 2

Offset: 0

Gregory L. Simay, Jun 14 2019

T(n,0) = A000007(n).
T(n,1) = A040000(n) for n > 0.
T(n,2) = A005843(n-1).
T(n,3) = 2*A007980(n-3).
T(n,4) = 2*A061866(n-1).
T(n,5) = 2*A091773(n-5).
Conjecture: T(n,k) = 2*(the associated Poincaré series). If T(n,1) were 1 for n>0, then T(n, k>1) would be a Poincaré series.

			T(5,3) = 8 and counts the overpartitions 3,1,1; 3',1,1; 3,1',1; 3',1',1; 2,2,1; 2',2,1; 2,2,1' and 2',2,1'.
T(16,5) = 404 = T(11,5) + 2*( T(11,4) + T(11,3) + T(11,2) + T(11,1)) = 72 + 2*(84 + 60 + 20 + 2) = 404.
T(16, 5) = T(15,4) + T(11,4) + T(10,4) + T(6,4) + T(5,4) = 248 + 84 + 60 + 8 + 4 = 404.
T(9,1) + T(8,2) + T(7,3) + T(6,4) + T(6,5)= 2 + 14 + 20 + 8 + 2 = 46 =A300415(10).
Triangle: T(n,k) begins:
  1;
  0, 2;
  0, 2,  2;
  0, 2,  4,  2;
  0, 2,  6,  4,  2;
  0, 2,  8,  8,  4,  2;
  0, 2, 10, 14,  8,  4,  2;
  0, 2, 12, 20, 16,  8,  4,  2;
  0, 2, 14, 28, 26, 16,  8,  4, 2;
  0, 2, 16, 38, 40, 28, 16,  8, 4, 2;
  0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2;
  ...

Row sums give A015128.
Main diagonal T(n,n) gives A040000.
Cf. A005843, A007980, A061866, A091773, A235792, A300415.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      expand(`if`(j>0, 2*x^j, 1)*b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jun 15 2019

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[If[j > 0, 2*x^j, 1]*b[n - i*j, i - 1]], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)

Sum_{k=0..n} T(n,k) = A015128(n), the number of overpartitions of n.
If k > n, T(n,k) = 0.
If n >= k > n/2, T(n,k) = 2*A015128(n-k).
Conjecture: T(n,k) = T(n-k, k) + 2*(T(n-k, k-1) + ... + T(n-k, 1)).
Conjecture: T(n,k) = T(n-1, k-1) + T(n-k, k-1) + T(n-k-1, k-1) + T(n-2k, k-1) + T(n-2k-1) + ...
Conjecture: T(n,1) + T(n-1,2) + ... + T(n-floor(n/2),floor(n/2)) = A300415(n+1).
T(n,2) = 2n - 2.
Conjecture: g.f. T(n,k) = 2*(1+x)(1+x^2)...(1+x^(k-1))/((1-x)...(1-x^k)).
Sum_{k=1..n} k * T(n,k) = A235792(n). - Alois P. Heinz, Jun 15 2019

A332735 Numbers of graphs which are double triangle descendants of K_5 with four more vertices than triangles.

Original entry on oeis.org

1, 6, 15, 34, 61, 106, 162, 246, 342, 477, 626, 825, 1039, 1314, 1606, 1970, 2352, 2817, 3302, 3881, 4481, 5186, 5914, 6758, 7626, 8621, 9642, 10801, 11987, 13322, 14686, 16210, 17764, 19489, 21246, 23185, 25157, 27322, 29522, 31926, 34366, 37021, 39714, 42633, 45591, 48786

Offset: 9

Karen A. Yeats, Feb 21 2020

See Laradji, Mishna, Yeats paper for definition of double triangle descendants.

Mohamed Laradji, Marni Mishna, and Karen Yeats, Some results on double triangle descendants of K_5, arXiv:1904.06923 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Double triangle descendants of K_5 with three more vertices than triangles is A007980. Double triangle descendants of K_5 with two more vertices than triangles is A008619. Double triangle descendants of K_5 with one more vertex than triangles is A000007. Double triangle descendants of K_5 with the same number of vertices as triangles is A000012.

G.f.: x^9*(1 + 4*x + 3*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 4*x^7 - 3*x^8 + 3*x^9 - x^10 + x^11)/((1 - x)^4*(1 + x)^2*(1 + x^2)). See Laradji, Mishna, Yeats paper for proof.

A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

Original entry on oeis.org

1, 7, 13, 25, 43, 61, 85, 115, 145, 181, 223, 265, 313, 367, 421, 481, 547, 613, 685, 763, 841, 925, 1015, 1105, 1201, 1303, 1405, 1513, 1627, 1741, 1861, 1987, 2113, 2245, 2383, 2521, 2665, 2815, 2965, 3121, 3283, 3445, 3613, 3787, 3961, 4141, 4327, 4513, 4705, 4903, 5101, 5305, 5515, 5725

Offset: 0

Aaron David Fairbanks, May 17 2023

In contrast, A003154 (the star numbers) counts the hexagonal lattice points within a hexagram centered at a lattice point and with the vertices of the central hexagon at the six lattice points a given number of steps outward from the central point.
Besides the first term, the first differences are given by six times A004396.
A005448 (the centered triangular numbers) counts just the lattice points within one of the two triangles that make up the hexagram.

			Illustration of initial terms:
.
.                                 o     o
.                   o   o          o o o
.         o o        o o          o o o o
.   o    o o o    o o o o o    o o o o o o o
.         o o        o o          o o o o
.                   o   o          o o o
.                                 o     o
.
.   1      7          13             25
.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A003154, A004396, A005448, A007980, A081272.

Table[6*Ceiling[n*(n + 1)/3] + 1, {n, 0, 60}] (* Amiram Eldar, Jul 28 2023 *)

a(n) = 6*ceil(n*(n+1)/3) + 1; \\ Michel Marcus, Jun 14 2024

a(n) = 6*ceiling(n*(n+1)/3) + 1.
a(n) = 6*A007980(n-1) + 1 for n >= 1.
a(n+1) - a(n) = 6*A004396(n+1).
a(3n) = A081272(n).
G.f.: (1 + 5*x + 5*x^3 + x^4)/((1 - x)^3*(1 + x + x^2)). - Stefano Spezia, Feb 06 2025

A383338 Square array read by antidiagonals, where the n-th row is the coordination sequence of a certain tiling with an n-dimensional analog of the X pentomino (or Greek cross), n >= 1.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 8, 8, 1, 2, 12, 26, 14, 1, 2, 16, 56, 76, 20, 1, 2, 20, 98, 244, 150, 28, 1, 2, 24, 152, 578, 632, 296, 38, 1, 2, 28, 218, 1138, 1882, 1680, 558, 48, 1, 2, 32, 296, 1984, 4492, 6424, 4336, 896, 60, 1, 2, 36, 386, 3176, 9230, 18908, 21782, 8688, 1422, 74, 1

Offset: 1

Pontus von Brömssen, Apr 29 2025

The tile consists of an n-dimensional central hypercube with one hypercube attached to each of its 2*n (n-1)-dimensional facets. n-dimensional space can be tiled with this tile by placing the centers of the tiles at integer points (x_1, ..., x_n) for which Sum_{j=1..n} j*x_j is divisible by 2*n+1. (See problem B6 of the 2019 Putnam competition). Two tiles are considered to be neighbors if they share an (n-1)-dimensional facet.

			Array begins:
  n\k| 0  1   2    3     4      5      6       7       8
  ---+--------------------------------------------------
  1  | 1  2   2    2     2      2      2       2       2
  2  | 1  4   8   12    16     20     24      28      32
  3  | 1  8  26   56    98    152    218     296     386
  4  | 1 14  76  244   578   1138   1984    3176    4774
  5  | 1 20 150  632  1882   4492   9230   17040   29042
  6  | 1 28 296 1680  6424  18908  46416   99904  194768
  7  | 1 38 558 4336 21782  80838 241730  616584 1393906
  8  | 1 48 896 8688 52896 232000 803232 2332896 5923776

Kiran S. Kedlaya, The 80th William Lowell Putnam Mathematical Competition, Dec 7 2019.
Kiran S. Kedlaya, Solutions to the 80th William Lowell Putnam Mathematical Competition, Dec 7 2019.
Yusuke Nakamura, Ryotaro Sakamoto, Takafumi Mase, and Junichi Nakagawa, Coordination sequences of crystals are of quasi-polynomial type, Acta Crystallographica A 77 (2021), 138-148.
Eric Weisstein's World of Mathematics, Greek Cross.
Index entries for sequences related to coordination sequences.

Rows: A040000 (n=1), A008574 (n=2), A005897 (n=3; empirically).

Cf. A007980.

T(n,0) = 1.
T(n,1) = 2*A007980(n-1).
T(1,k) = A040000(k).
T(2,k) = A008574(k).
Empirically (do these formulas follow from the results of Nakamura et al.?):
  T(3,k) = A005897(k).
  T(4,k) = 10*k^3 - 7*k^2 + 13*k - 2 for k >= 1.
  T(5,k) = (22/3)*k^4 - 4*k^3 + (50/3)*k^2 - 2*k + 2 for k >= 1.
  T(6,k) = (32/5)*k^5 - 7*k^4 + 28*k^3 - 11*k^2 + (58/5)*k for k >= 1.
  T(7,k) = (304/45)*k^6 - (284/15)*k^5 + (1237/18)*k^4 - 86*k^3 + (8777/90)*k^2 - (601/15)*k + 10 for k >= 1.

cp's OEIS Frontend

A260253 Number of symmetry-allowed, linearly-independent terms at n-th order in the expansion of E x (e+a) rovibrational perturbation matrix H(Jx,Jy,Jz).

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A347026 Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.

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C

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A115281 Correlation triangle for the sequence 2-0^n.

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A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n.

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A332735 Numbers of graphs which are double triangle descendants of K_5 with four more vertices than triangles.

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A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

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A383338 Square array read by antidiagonals, where the n-th row is the coordination sequence of a certain tiling with an n-dimensional analog of the X pentomino (or Greek cross), n >= 1.

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