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A001251
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Number of permutations of order n with the length of longest run equal to 3.
(Formerly M2031 N0803)
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12
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0, 0, 2, 12, 70, 442, 3108, 24216, 208586, 1972904, 20373338, 228346522, 2763212980, 35926266244, 499676669254, 7405014187564, 116511984902094, 1940073930857802, 34087525861589564, 630296344519286304, 12235215845125112122, 248789737587365945992
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OFFSET
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1,3
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REFERENCES
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F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Terms for n>=13 are incorrect.)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) ~ c * d^n * n!, where d = 0.92403585760753647721113386869798700855648617941... is the root of the equation 8 - 2*sin(sqrt(phi)/d) * (2*sqrt(5*(phi-1)) * cosh(sqrt(phi-1)/d) + 2*sinh(sqrt(phi-1)/d)) + 2*cos(sqrt(phi)/d) * (6*cosh(sqrt(phi-1)/d) + 2*sqrt(5*phi) * sinh(sqrt(phi-1)/d)) = 0, phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 1.259371257828351725264434486385284120241474052544197367866029465830756911... - Vaclav Kotesovec, Sep 06 2014, updated Aug 18 2018
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MATHEMATICA
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length = 3;
g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[t<length, Sum[g[u + j - 1, o - j, t+1], {j, 1, o}], 0]];
b[u_, o_, t_] := b[u, o, t] = If[t == length, g[u, o, t], Sum[b[o + j - 1, u - j, 2], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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