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A245518
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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.
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0
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1, 0, 1, 4, 2, 2, 4, 16, 12, 8, 12, 16, 64, 64, 40, 40, 64, 64, 284, 328, 236, 176, 236, 328, 284, 1360, 1760, 1432, 1000, 1000, 1432, 1760, 1360, 7184, 9928, 9092, 6536, 5312, 6536, 9092, 9928, 7184
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OFFSET
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4,4
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LINKS
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FORMULA
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a(n,i) = sum_{L=1..^n-2} product_{k=1..n} d(L,k,i), where
for i < L,
d(L,k) if 1 <= k <= i,
d(L,k,i) ={ d(L,k) - 1 if i < k < n - i,
d(L,k) if n - i <= k <= n;
for i > L + 1,
d(L,k) if 1 <= k <= n - i,
d(L,k,i) ={ d(L,k) - 1 if n - i < k < n - i + L + 2,
d(L,k) if n - i + L + 2 <= k <= n.
k if 1 <= k < m,
d(L,k) ={ L + 1 if m <= k <= M,
n + 1 - k if M < k <= n,
m = min{L + 1, n - L}, M = max{L + 1, n - L}.
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EXAMPLE
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For n=4 and i=2, a(4,2) = 0.
For n=8 and i=5, a(8,5) = 64.
Triangle begins:
[n\i] [1] [2] [3] [4] [5] [6] [7] [8] [9]
[4] 1, 0, 1;
[5] 4, 2, 2, 4;
[6] 16, 12, 8, 12, 16;
[7] 64, 64, 40, 40, 64, 64;
[8] 284, 328, 236, 176, 236, 328, 284;
[9] 1360, 1760, 1432, 1000, 1000, 1432, 1760, 1360;
[10] 7184, 9928, 9092, 6536, 5312, 6536, 9092, 9928, 7184;
. . .
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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STATUS
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approved
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