A050447 Table T(n,m) giving total degree of n-th-order elementary symmetric polynomials in m variables, -1 <= n, 1 <= m, transposed and read by upward antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 14, 8, 1, 1, 6, 15, 30, 31, 13, 1, 1, 7, 21, 55, 85, 70, 21, 1, 1, 8, 28, 91, 190, 246, 157, 34, 1, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 1, 11, 55, 285, 1086, 3164, 6405, 8272, 5864, 1782, 144, 1

Offset

0,5

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 (see p. 120).

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.

Links

T. D. Noe, Table of 100 antidiagonals

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]

G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973.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

Formula

See PARI code. See A050446 for recurrence.

G.f. for row n >= 0: f(n, x) = (x + f(n-2, x))/(1 - x^2 - x*f(n-2, x)), where f(0, x) = 1 and f(1, x) = 1/(1 - x) [R. P. Stanley]. - L. Edson Jeffery, Oct 19 2017

Example

Table begins
.    1   1   1    1     1      1       1       1        1         1
.    1   2   3    5     8     13      21      34       55        89
.    1   3   6   14    31     70     157     353      793      1782
.    1   4  10   30    85    246     707    2037     5864     16886
.    1   5  15   55   190    671    2353    8272    29056    102091
.    1   6  21   91   371   1547    6405   26585   110254    457379
.    1   7  28  140   658   3164   15106   72302   345775   1654092
.    1   8  36  204  1086   5916   31998  173502   940005   5094220
.    1   9  45  285  1695  10317   62349  377739  2286648  13846117
.    1  10  55  385  2530  17017  113641  760804  5089282  34053437

Mathematica

nmax = 12; 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[k-1, n-k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011 *)
nmax = 10; f[0, x_] := 1; f[1, x_] := 1/(1 - x); f[n_, x_] := (x + f[n - 2, x])/(1 - x^2 - x*f[n - 2, x]); t[n_, m_] := Coefficient[Series[f[n, x], {x, 0, m}], x, m]; Grid[Table[t[n, m], {n, nmax}, {m, 0, nmax - 1}]] (* L. Edson Jeffery, Oct 19 2017 *)

Prog

(PARI) M(n)=matrix(n, n, i, j, if(sign(i+j-n)-1, 0, 1)); V(n)=vector(n, i, 1); P(r, n)=vecmax(V(r)*M(r)^n) \\ P(r, n) is T(n, k); Benoit Cloitre, Jan 27 2003

Crossrefs

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

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

Cf. A001924, A050446.

Sequence in context: A171848 A144151 A022818 * A248601 A167172 A173075

Adjacent sequences: A050444 A050445 A050446 * A050448 A050449 A050450

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane, Dec 23 1999

Extensions

More terms from Naohiro Nomoto, Jul 03 2001

Status

approved