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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.

%I A050447 #48 Jul 11 2023 12:38:22
%S A050447 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,
%T A050447 55,85,70,21,1,1,8,28,91,190,246,157,34,1,1,9,36,140,371,671,707,353,
%U A050447 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
%N 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.
%D A050447 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D A050447 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
%D A050447 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.
%H A050447 T. D. Noe, <a href="/A050447/b050447.txt">Table of 100 antidiagonals</a>
%H A050447 J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H A050447 G. Kreweras, <a href="http://www.numdam.org/item?id=MSH_1976__53__5_0">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H A050447 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.
%F A050447 See PARI code. See A050446 for recurrence.
%F A050447 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
%e A050447 Table begins
%e A050447 .    1   1   1    1     1      1       1       1        1         1
%e A050447 .    1   2   3    5     8     13      21      34       55        89
%e A050447 .    1   3   6   14    31     70     157     353      793      1782
%e A050447 .    1   4  10   30    85    246     707    2037     5864     16886
%e A050447 .    1   5  15   55   190    671    2353    8272    29056    102091
%e A050447 .    1   6  21   91   371   1547    6405   26585   110254    457379
%e A050447 .    1   7  28  140   658   3164   15106   72302   345775   1654092
%e A050447 .    1   8  36  204  1086   5916   31998  173502   940005   5094220
%e A050447 .    1   9  45  285  1695  10317   62349  377739  2286648  13846117
%e A050447 .    1  10  55  385  2530  17017  113641  760804  5089282  34053437
%t A050447 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 *)
%t A050447 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 *)
%o A050447 (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
%Y A050447 Rows give A000012, A000045, A006356, A006357, A006358, ...
%Y A050447 Columns give A000012, A000027, A000217, A000330, A006322, ...
%Y A050447 Cf. A001924, A050446.
%K A050447 nonn,easy,nice,tabl
%O A050447 0,5
%A A050447 _N. J. A. Sloane_, Dec 23 1999
%E A050447 More terms from _Naohiro Nomoto_, Jul 03 2001