A293310 -id:A293310 - cp's OEIS frontend

A006325 4-dimensional analog of centered polygonal numbers.

0, 0, 1, 7, 26, 70, 155, 301, 532, 876, 1365, 2035, 2926, 4082, 5551, 7385, 9640, 12376, 15657, 19551, 24130, 29470, 35651, 42757, 50876, 60100, 70525, 82251, 95382, 110026, 126295, 144305, 164176, 186032, 210001, 236215, 264810, 295926

Offset: 0

Albert Rich (Albert_Rich(AT)msn.com)

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
For n>0, a(n+1) is the n-th antidiagonal sum of A213751. - Clark Kimberling, Jun 20 2012
This sequence is the case m=n-1, k=n+3 of b(m,k) = m*(m+1)*((k-2)*m-(k-5))/6, which is the m-th k-gonal pyramidal number. - Luciano Ancora, Apr 11 2015
Starting (1, 7, 26, ...), this is the binomial transform of (1, 6, 13, 12, 4, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
Also starting (1, 7, 26, ...), this appears to be the number of magic labelings of the cycle-of-loops graph LOOP X C_4 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. - David J. Seal, Sep 13 2017
The conjecture by David J. Seal is true and easily proved using MacMahon's Omega operators via the "Omega" package for Mathematica authored by Axel Riese (obtaining (up to an offset) the generating function listed in the formula section below). See the second Mathematica program in which the edges of LOOP X C_4 are indexed as in the example below. The Omega package can be downloaded from the link provided in the article by G. E. Andrews et al. - L. Edson Jeffery, Oct 15 2017

			A representation of the LOOP X C_4 graph, with edges and loops indexed as shown, as used in the second Mathematica program below:
.             3         1
.              O_______O
.              |   2   |
.              |4     0|
.              |_______|
.              O   6   O
.             5         7

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
Richard P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005900, A169938, A213751, A288486, A019298, A061907, A244497, A244879, A244873, A244880.
Cf. A000027, A000217, A019298, A244497, A244879, A244873, A244880, A293310, A293309 (magic labelings of LOOP X C_k, for k = 1..3,5..10).
Cf. A293311, A293312.

[n*(n-1)*(n^2-n+1)/6: n in [0..40]]; // Vincenzo Librandi, May 22 2011

Table[n*(n-1)*(n^2-n+1)/6, {n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
<< Omega.m; n = 4; cond = {}; Do[AppendTo[cond, Sum[a[Mod[2*k - j, 2*n]], {j, 0, 2}] == a[2*n]], {k, 0, n - 1}]; f = OEqSum[Product[x[i]^a[i], {i, 0, 2*n}], cond, u][[1]]; Do[f = OEqR[f, Subscript[u, k]], {k, n}];
(* Generating function: *)
f = Factor[f /. {x[2*n] -> x} /. {x[_] -> 1}]
(* This sequence (with initial zeros dropped): *)
CoefficientList[Series[f, {x, 0, 35}], x] (* L. Edson Jeffery, Oct 15 2017 *)

a(n)=n*(n-1)*(n^2-n+1)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(n-1)*(n^2-n+1)/6.
a(n) = ((n^5 - (n-1)^5) - (n^1 - (n-1)^1))/30 = (n^5 - (n-1)^5 - 1)/30. - Xavier Acloque, Jan 25 2003
The partial sums of the octahedral numbers: a(n+1) = Sum_{i=0..n} A005900(i). - Jonathan Vos Post, Mar 14 2006
G.f.: -x^2*(x+1)^2/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013 [Which is just rephrasing the partial sum formula with the Murthy formula in A005900. - R. J. Mathar, Jun 14 2014]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Yosu Yurramendi, Sep 03 2013
From Daniel Poveda Parrilla, Sep 09 2017: (Start)
a(n) = A169938(n-1)/6.
a(n+1) = A288486(n)/24. (End)
Sum_{n>=2} 1/a(n) = 12 - 2 * sqrt(3) * tanh(sqrt(3)*Pi/2). - Amiram Eldar, Jun 28 2020
E.g.f.: exp(x)*x^2*(3 + 4*x + x^2)/6. - Stefano Spezia, Dec 12 2021

A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 11, 7, 1, 6, 15, 23, 26, 11, 1, 7, 21, 42, 70, 57, 18, 1, 8, 28, 69, 155, 197, 129, 29, 1, 9, 36, 106, 301, 533, 571, 289, 47, 1, 10, 45, 154, 532, 1223, 1884, 1640, 650, 76, 1, 11, 55, 215, 876, 2494, 5103, 6604, 4726, 1460, 123, 1

Offset: 1

L. Edson Jeffery, Oct 06 2017

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for row n of this array was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews, et al., and is needed for the Mathematica program below.

			Array begins:
.  1   2    3     4      5       6       7        8        9        10
.  1   3    6    10     15      21      28       36       45        55
.  1   4   11    23     42      69     106      154      215       290
.  1   7   26    70    155     301     532      876     1365      2035
.  1  11   57   197    533    1223    2494     4654     8105     13355
.  1  18  129   571   1884    5103   11998    25362    49347     89848
.  1  29  289  1640   6604   21122   57271   137155   298184    599954
.  1  47  650  4726  23219   87677  274132   743724  1806597   4016683
.  1  76 1460 13604  81555  363606 1310974  4029310 10936124  26868719
.  1 123 3281 39175 286555 1508401 6271378 21836366 66220705 179784715

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A293311.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (rows 1..10).

(* Run this first: *)
<< Omega.m;
(* Then run the following in a different cell: *)
nmax = 11; Do[cond = {}; Do[If[n == 1, AppendTo[cond, Subscript[a, 1] + Subscript[a, 2] == Subscript[a, 3]]; Break[], AppendTo[cond, If[j == n, Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 1] == Subscript[a, 2*n + 1], Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 2*j + 1] == Subscript[a, 2*n + 1]]]], {j, n}]; f = OEqSum[Product[Subscript[x, i]^Subscript[a, i], {i, 2*n + 1}], cond, \[Lambda]][[1]] /. {Subscript[x, 2*n + 1] -> z} /. {Subscript[x, _] -> 1}; Do[f = OEqR[f, Subscript[\[Lambda], k]], {k, Length[cond]}]; Do[a293311[n, k] = Coefficient[Series[f, {z, 0, nmax - 1}], z, k], {k, 0, nmax - 1}], {n, nmax}];
(* Array: *)
Grid[Table[a293311[n, k], {n, nmax}, {k, 0, nmax - 1}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a293311[n, k - n], {k, 11}, {n, k}]]

A293309 Number of magic labelings of the graph LOOP X C_10 (see comments) having magic sum n, n >= 0.

Original entry on oeis.org

1, 123, 3281, 39175, 286555, 1508401, 6271378, 21836366, 66220705, 179784715, 445824731, 1025102013, 2211041131, 4514532465, 8789910980, 16416797116, 29556115153, 51502789451, 87162399205, 143684487475, 231291309931, 364347612673, 562724586326

Offset: 0

L. Edson Jeffery, Oct 05 2017

nonn

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for this sequence was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews et al.

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A293311, A293312.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310 (magic labelings of LOOP X C_k, for k=1..9).

CoefficientList[Series[(1 + 112*z + 1983*z^2 + 9684*z^3 + 16120*z^4 + 9684*z^5 + 1983*z^6 + 112*z^7 + z^8)/(1 - z)^11, {z, 0, 22}], z]
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 123, 3281, 39175, 286555, 1508401, 6271378, 21836366, 66220705, 179784715, 445824731}, 25] (* Vincenzo Librandi, Oct 12 2017 *)

G.f.: (1 + 112*z + 1983*z^2 + 9684*z^3 + 16120*z^4 + 9684*z^5 + 1983*z^6 + 112*z^7 + z^8)/(1 - z)^11.

A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 4, 6, 2, 5, 1, 7, 11, 10, 3, 6, 1, 11, 26, 23, 15, 3, 7, 1, 18, 57, 70, 42, 21, 4, 8, 1, 29, 129, 197, 155, 69, 28, 4, 9, 1, 47, 289, 571, 533, 301, 106, 36, 5, 10, 1, 76, 650, 1640, 1884, 1223, 532, 154, 45, 5, 11

Offset: 1

L. Edson Jeffery, Oct 10 2017

Conjecture: For all n >= 1, for all k >= 2, A(n, k) = A293311(k, n); i.e., A(n, k) = number of magic labelings of the graph LOOP X C_k with magic sum n - 1.

			Array begins:
.   1 1  1   1    1     1      1       1       1        1         1
.   2 1  3   4    7    11     18      29      47       76       123
.   3 2  6  11   26    57    129     289     650     1460      3281
.   4 2 10  23   70   197    571    1640    4726    13604     39175
.   5 3 15  42  155   533   1884    6604   23219    81555    286555
.   6 3 21  69  301  1223   5103   21122   87677   363606   1508401
.   7 4 28 106  532  2494  11998   57271  274132  1310974   6271378
.   8 4 36 154  876  4654  25362  137155  743724  4029310  21836366
.   9 5 45 215 1365  8105  49347  298184 1806597 10936124  66220705
.  10 5 55 290 2035 13355  89848  599954 4016683 26868719 179784715
.  11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731
.  ...

Cf. A293311.
Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).
Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).

s[0, x_] := 1; s[1, x_] := x; s[k_, x_] := x*s[k - 1, x] - s[k - 2, x]; c[n_, j_] := 2 (-1)^(j - 1) Cos[j*Pi/(2 n + 1)]; a[n_, k_] := Round[Sum[s[n - 1, c[n, j]]^(k), {j, n}]];
(* Array: *)
Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]

Let S(0, x) = 1, S(1, x) = x, S(k, x) = x*S(k - 1, x) - S(k - 2, x) (the S-polynomials of Wolfdieter Lang) and c(n, j) = 2*(-1)^(j - 1)*cos(j*Pi/(2*n + 1)). Then A(n, k) = Sum_{j=1..n} S(n - 1, c(n, j))^(k), n >= 1, k >= 0.

cp's OEIS Frontend

A006325 4-dimensional analog of centered polygonal numbers.

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A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.

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A293309 Number of magic labelings of the graph LOOP X C_10 (see comments) having magic sum n, n >= 0.

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A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

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