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%I #48 Jul 11 2023 12:38:22
%S 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 55,85,70,21,1,1,8,28,91,190,246,157,34,1,1,9,36,140,371,671,707,353,
%U 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 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 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D 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 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 T. D. Noe, <a href="/A050447/b050447.txt">Table of 100 antidiagonals</a>
%H 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 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 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.
%F See PARI code. See A050446 for recurrence.
%F 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 Table begins
%e . 1 1 1 1 1 1 1 1 1 1
%e . 1 2 3 5 8 13 21 34 55 89
%e . 1 3 6 14 31 70 157 353 793 1782
%e . 1 4 10 30 85 246 707 2037 5864 16886
%e . 1 5 15 55 190 671 2353 8272 29056 102091
%e . 1 6 21 91 371 1547 6405 26585 110254 457379
%e . 1 7 28 140 658 3164 15106 72302 345775 1654092
%e . 1 8 36 204 1086 5916 31998 173502 940005 5094220
%e . 1 9 45 285 1695 10317 62349 377739 2286648 13846117
%e . 1 10 55 385 2530 17017 113641 760804 5089282 34053437
%t 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 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 (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 Rows give A000012, A000045, A006356, A006357, A006358, ...
%Y Columns give A000012, A000027, A000217, A000330, A006322, ...
%Y Cf. A001924, A050446.
%K nonn,easy,nice,tabl
%O 0,5
%A _N. J. A. Sloane_, Dec 23 1999
%E More terms from _Naohiro Nomoto_, Jul 03 2001
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