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A050446
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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, read by upward antidiagonals.
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19
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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
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OFFSET
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0,5
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COMMENTS
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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
Let LOOP X C_k, k >= 1, be the graph constructed by attaching a loop to each vertex of the cycle graph C_k. Let G_n, n >= 0, be the graph obtained by deleting one edge from LOOP X C_{n+1} while retaining the n + 1 loops; e.g., for n = 4, see the graph G_4 at the top of the page in the Stanley link below. Then T(n,m) equals the number of magic labelings of G_n having magic sum m. (See the second Mathematica program below which requires the "Omega" package authored by Axel Riese and which can be downloaded from the link provided in the article by Andrews et al.) - L. Edson Jeffery, Oct 19 2017
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REFERENCES
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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).
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LINKS
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FORMULA
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T(n, m) = T(n, m-1) + Sum( T(2k, m-1)*T(n-1-2k, m), {k, 0, (n-1)/2}).
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EXAMPLE
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Table begins
. 1 1 1 1 1 1 1 1 1 1
. 1 2 3 4 5 6 7 8 9 10
. 1 3 6 10 15 21 28 36 45 55
. 1 5 14 30 55 91 140 204 285 385
. 1 8 31 85 190 371 658 1086 1695 2530
. 1 13 70 246 671 1547 3164 5916 10317 17017
. 1 21 157 707 2353 6405 15106 31998 62349 113641
. 1 34 353 2037 8272 26585 72302 173502 377739 760804
. 1 55 793 5864 29056 110254 345775 940005 2286648 5089282
. 1 89 1782 16886 102091 457379 1654092 5094220 13846117 34053437
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MAPLE
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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
end do:
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MATHEMATICA
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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 *)
<< Omega.m; nmax = 9; Do[cond[n_] = {}; If[n == 0, cond[n] = {a[1] == a[2]}, AppendTo[cond[n], {a[1] + a[2] == a[2 n + 2], a[2 n] + a[2 n + 1] == a[2 n + 2]}]; If[n > 1, Do[AppendTo[cond[n], a[2 j] + a[2 j + 1] + a[2 j + 2] == a[2 n + 2]], {j, n - 1}]]]; cond[n] = Flatten[cond[n]]; f[n_] = OEqSum[Product[x[i]^a[i], {i, 2 n + 2}], cond[n], u][[1]] /. x[2 n + 2] -> y /. x[_] -> 1; Do[f[n] = OEqR[f[n], Subscript[u, j]], {j, Length[cond[n]]}], {n, 0, nmax}]; Grid[Table[CoefficientList[Series[f[n], {y, 0, nmax}], y], {n, 0, nmax}]] (* L. Edson Jeffery, Oct 19 2017 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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