OFFSET
0,5
COMMENTS
T(n, m) is a polynomial of degree n in m. For example, T(2, m) = (m + 1)(m + 2)/2. For the polynomials corresponding to n = 1, 2, ..., 10, see the Cyvin-Gutman reference (p. 143). Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005
Let LOOP X C_k, k >= 1, be the graph constructed by attaching a loop to each vertex of the cycle graph C_k. Let G_n, n >= 0, be the graph obtained by deleting one edge from LOOP X C_{n+1} while retaining the n + 1 loops; e.g., for n = 4, see the graph G_4 at the top of the page in the Stanley link below. Then T(n, m) equals the number of magic labelings of G_n having magic sum m. (See the second Mathematica program below which requires the "Omega" package authored by Axel Riese and which can be downloaded from the link provided in the article by Andrews et al.) - L. Edson Jeffery, Oct 19 2017
REFERENCES
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 142-144).
LINKS
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Byeong-Gil Choe and Hyeong-Kwan Ju, A Recurrence Relation Associated with Unit-Primitive Matrices, arXiv:2305.03930 [math.CO], 2023.
Jane Ivy Coons and Seth Sullivant, The h*-polynomial of the order polytope of the zig-zag poset, arXiv:1901.07443 [math.CO], 2019.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017).
Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.
Guoce Xin and Yueming Zhong, Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials, arXiv:2201.02376 [math.CO], 2022.
FORMULA
T(n, m) = T(n, m - 1) + Sum_{k=0..(n-1)/2} T(2*k, m - 1)*T(n - 1 - 2*k, m).
EXAMPLE
Array begins:
[0] 1 1 1 1 1 1 1 1 1 1
[1] 1 2 3 4 5 6 7 8 9 10
[2] 1 3 6 10 15 21 28 36 45 55
[3] 1 5 14 30 55 91 140 204 285 385
[4] 1 8 31 85 190 371 658 1086 1695 2530
[5] 1 13 70 246 671 1547 3164 5916 10317 17017
[6] 1 21 157 707 2353 6405 15106 31998 62349 113641
[7] 1 34 353 2037 8272 26585 72302 173502 377739 760804
[8] 1 55 793 5864 29056 110254 345775 940005 2286648 5089282
[9] 1 89 1782 16886 102091 457379 1654092 5094220 13846117 34053437
...
Triangle starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 1;
[3] 1, 3, 3, 1;
[4] 1, 5, 6, 4, 1;
[5] 1, 8, 14, 10, 5, 1;
[6] 1, 13, 31, 30, 15, 6, 1;
MAPLE
A050446 := proc(n, m)
option remember;
if m=0 then
1;
else
procname(n, m-1)+add( procname(2*k, m-1) *procname(n-1-2*k, m), k=0..floor((n-1)/2) );
end if;
end proc:
for d from 0 to 12 do
for m from 0 to d do
printf("%d, ", A050446(d-m, m)) ;
end do:
end do: # R. J. Mathar, Dec 14 2011
MATHEMATICA
t[n_, m_?Positive] := t[n, m] = t[n, m-1] + Sum[t[2k, m-1]*t[n-1 - 2k, m], {k, 0, (n-1)/2}]; t[n_, 0] = 1; Flatten[Table[t[i-k , k-1], {i, 1, 12}, {k, 1, i}]] (* Jean-François Alcover, Jul 25 2011, after formula *)
<< Omega.m; nmax = 9; Do[cond[n_] = {}; If[n == 0, cond[n] = {a[1] == a[2]}, AppendTo[cond[n], {a[1] + a[2] == a[2 n + 2], a[2 n] + a[2 n + 1] == a[2 n + 2]}]; If[n > 1, Do[AppendTo[cond[n], a[2 j] + a[2 j + 1] + a[2 j + 2] == a[2 n + 2]], {j, n - 1}]]]; cond[n] = Flatten[cond[n]]; f[n_] = OEqSum[Product[x[i]^a[i], {i, 2 n + 2}], cond[n], u][[1]] /. x[2 n + 2] -> y /. x[_] -> 1; Do[f[n] = OEqR[f[n], Subscript[u, j]], {j, Length[cond[n]]}], {n, 0, nmax}]; Grid[Table[CoefficientList[Series[f[n], {y, 0, nmax}], y], {n, 0, nmax}]] (* L. Edson Jeffery, Oct 19 2017 *)
PROG
(Python)
from functools import cache
@cache
def T(n, k):
return T(n, k - 1) + sum(T(2 * j, k - 1) * T(n - 1 - 2 * j, k)
for j in range(1 + (n - 1) // 2)) if k > 0 else 1
for n in range(6): print([T(n - k, k) for k in range(n + 1)])
# Peter Luschny, Jun 08 2024
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Dec 23 1999
EXTENSIONS
More terms from Naohiro Nomoto, Jul 03 2001
STATUS
approved