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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 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

For the rows and columns of the table in the example below, see the comment by Jeffery in A050447. - L. Edson Jeffery, Dec 15 2011

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.

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

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 2 3 4 5 6 7 ...

1 3 6 10 15 21 28 ...

1 5 14 30 55 91 140 ...

1 8 31 85 190 371 658 ...

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 *)

CROSSREFS

Rows give A000217, A000330, A006322, ...

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

Cf. A050447.

Sequence in context: A026736 A230859 A213086 * A214868 A144048 A258708

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 May 30 03:33 EDT 2017. Contains 287305 sequences.