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A050447
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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 antidiagonals.
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17
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| From L. Edson Jeffery, Nov 28, 2011: (Start) For the table T in the example below, and for n=1,2,..., k=1,2,..., the n-th column is generated by the n X n matrix M_n with entries [M_n]_(i,j) = if(j>=n-i+1,1,0), i=1,...,n, j=1,...,n, by {T(n,k)}={[[M_n]^(k+2)]_(1,1)}. Thus M_n is a unit-primitive matrix (see [Jeffery]) of the form M_n = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1]. The generating functions for the columns are of the form F_n(x) = G_n(x)/H_n(x) for which the coefficients in the denominators are given by row n of A187660. The n-th row of the table has generating function of the form f_n(x) = B_n(x)/(1-x)^n, in which B_1(x) = B_2(x) = 1 and, for n >= 3, the coefficients for B_n(x) are given by the (n-2)-th row of the triangle:
1
1,1
1,3,1
1,7,7,1
1,14,31,14,1
1,26,109,109,26,1
etc., in which the second column is conjectured to be essentially A001924 = {1, 3, 7, 14, 26, 46, 79, 133, 221,...}. For example, row 8 of the table is the sequence {1,34,464,2037,8272,26585,72302,173502,377739,...} with generating function = (1+26*x+109*x^2+109*x^3+26*x^4+x^5)/(1-x)^8. (End)
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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, Kekule 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.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
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LINKS
| L. E. Jeffery, Unit-primitive matrices
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FORMULA
| See PARI code. See A050446 for recurrence.
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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 ...
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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}]] (* From Jean-François Alcover, Nov 14 2011 *)
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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) (from Benoit Cloitre, Jan 27, 2003. P(r, n) is T(n, k).)
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CROSSREFS
| Rows give A000217, A000330, A006322, ..., columns give A000045, A006356, A006357, A006358, ... Cf. A001924, A050446, A187660.
Sequence in context: A171848 A144151 A022818 * A167172 A173075 A166293
Adjacent sequences: A050444 A050445 A050446 * A050448 A050449 A050450
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KEYWORD
| nonn,easy,nice,tabl,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 23 1999
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EXTENSIONS
| More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jul 03 2001
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