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A245518 Irregular triangle read by rows: T(n,i) = number of alpha-labeled graphs with n edges that do not use the label i, for 1 <= i <= n-1 and n >= 4.

%I A245518 #22 Nov 22 2017 02:20:00
%S A245518 1,0,1,4,2,2,4,16,12,8,12,16,64,64,40,40,64,64,284,328,236,176,236,
%T A245518 328,284,1360,1760,1432,1000,1000,1432,1760,1360,7184,9928,9092,6536,
%U A245518 5312,6536,9092,9928,7184
%N A245518 Irregular triangle read by rows: T(n,i) = number of alpha-labeled graphs with n edges that do not use the label i, for 1 <= i <= n-1 and n >= 4.
%H A245518 Christian Barrientos, Sarah Minion, <a href="https://dx.doi.org/10.7151/dmgt.1985">On the number of alpha-labeled graphs</a>, Discussiones Mathematicae Graph Theory, to appear.
%H A245518 J. A. Gallian, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6">A dynamic survey of graph labeling</a>, Elec. J. Combin., (2013), #DS6.
%H A245518 David A. Sheppard, <a href="http://dx.doi.org/10.1016/0012-365X(76)90051-0">The factorial representation of major balanced labelled graphs</a>, Discrete Math., 15(1976), no. 4, 379-388.
%F A245518 a(n,i) = sum_{L=1..^n-2} product_{k=1..n} d(L,k,i), where
%F A245518 for i < L,
%F A245518             d(L,k)       if 1 <= k <= i,
%F A245518 d(L,k,i) ={ d(L,k) - 1   if i < k < n - i,
%F A245518             d(L,k)       if n - i <= k <= n;
%F A245518 for i > L + 1,
%F A245518             d(L,k)       if 1 <= k <= n - i,
%F A245518 d(L,k,i) ={ d(L,k) - 1   if n - i < k < n - i + L + 2,
%F A245518             d(L,k)       if n - i + L + 2 <= k <= n.
%F A245518           k          if 1 <= k < m,
%F A245518 d(L,k) ={ L + 1      if m <= k <= M,
%F A245518           n + 1 - k  if M < k <= n,
%F A245518 m = min{L + 1, n - L}, M = max{L + 1, n - L}.
%e A245518 For n=4 and i=2, a(4,2) = 0.
%e A245518 For n=8 and i=5, a(8,5) = 64.
%e A245518 Triangle begins:
%e A245518 [n\i] [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]     [9]
%e A245518 [4]    1,      0,      1;
%e A245518 [5]    4,      2,      2,      4;
%e A245518 [6]    16,     12,     8,      12,     16;
%e A245518 [7]    64,     64,     40,     40,     64,     64;
%e A245518 [8]    284,    328,    236,    176,    236,    328,    284;
%e A245518 [9]    1360,   1760,   1432,   1000,   1000,   1432,   1760,   1360;
%e A245518 [10]   7184,   9928,   9092,   6536,   5312,   6536,   9092,   9928,   7184;
%e A245518 . . .
%Y A245518 Cf. A241094, A005193.
%K A245518 easy,nonn,tabf
%O A245518 4,4
%A A245518 _Sarah Minion_ and _Christian Barrientos_, Jul 24 2014