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A168194
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a(n) = n^4*(n^3 + 1)/2.
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2
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0, 1, 72, 1134, 8320, 39375, 140616, 412972, 1050624, 2394765, 5005000, 9750906, 17926272, 31388539, 52725960, 85455000, 134250496, 205211097, 306162504, 447001030, 640080000, 900641511, 1247296072, 1702552644, 2293401600
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OFFSET
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0,3
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COMMENTS
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Number of unoriented rows of length 7 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=72, there are 2^7=128 oriented arrangements of two colors. Of these, 2^4=16 are achiral. That leaves (128-16)/2=56 chiral pairs. Adding achiral and chiral, we get 72. - Robert A. Russell, Nov 13 2018
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LINKS
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FORMULA
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G.f.: x*(1 + 64*x + 586*x^2 + 1208*x^3 + 605*x^4 + 56*x^5)/(1-x)^8. - Colin Barker, Apr 26 2012
G.f.: (Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..6} A145882(7,k) * x^k / (1-x)^8.
E.g.f.: (Sum_{k=1..7} S2(7,k)*x^k + Sum_{k=1..4} S2(4,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>7, a(n) = Sum_{j=1..8} -binomial(j-9,j) * a(n-j). (End)
E.g.f.: x*(2 +70*x +307*x^2 +351*x^3 +140*x^4 +21*x^5 +x^6)*exp(x)/2. - G. C. Greubel, Nov 14 2018
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MATHEMATICA
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PROG
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(PARI) vector(50, n, n--; n^4*(n^3+1)/2) \\ G. C. Greubel, Nov 14 2018
(Sage) [n^4*(n^3+1)/2 for n in (0..50)] # G. C. Greubel, Nov 14 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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