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A050446 Table T(n,m) giving total degree of n-th-order elementary symmetric polynomials in m variables, -1 <= n, 1 <= m, read by upward antidiagonals. 19
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 8, 14, 10, 5, 1, 1, 13, 31, 30, 15, 6, 1, 1, 21, 70, 85, 55, 21, 7, 1, 1, 34, 157, 246, 190, 91, 28, 8, 1, 1, 55, 353, 707, 671, 371, 140, 36, 9, 1, 1, 89, 793, 2037, 2353, 1547, 658, 204, 45, 10, 1, 1, 144, 1782, 5864, 8272, 6405, 3164, 1086, 285, 55, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..77.

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]

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.

Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658, 2015

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.

FORMULA

T(n, m) = T(n, m-1) + Sum( T(2k, m-1)*T(n-1-2k, m), {k, 0, (n-1)/2}).

EXAMPLE

Table begins

.    1  1    1     1      1      1       1       1        1        1

.    1  2    3     4      5      6       7       8        9       10

.    1  3    6    10     15     21      28      36       45       55

.    1  5   14    30     55     91     140     204      285      385

.    1  8   31    85    190    371     658    1086     1695     2530

.    1 13   70   246    671   1547    3164    5916    10317    17017

.    1 21  157   707   2353   6405   15106   31998    62349   113641

.    1 34  353  2037   8272  26585   72302  173502   377739   760804

.    1 55  793  5864  29056 110254  345775  940005  2286648  5089282

.    1 89 1782 16886 102091 457379 1654092 5094220 13846117 34053437

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 *)

CROSSREFS

Rows give A000012, A000027, A000217, A000330, A006322, ...

Columns give A000012, A000045, A000045, A006356, A006357, A006358, ...

Cf. A050447.

Sequence in context: A026736 A230859 A213086 * A214868 A144048 A292193

Adjacent sequences:  A050443 A050444 A050445 * A050447 A050448 A050449

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane, Dec 23 1999

EXTENSIONS

More terms from Naohiro Nomoto, Jul 03 2001

STATUS

approved

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Last modified November 16 13:54 EST 2018. Contains 317274 sequences. (Running on oeis4.)