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A036075
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The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.
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6
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1, 2, 10, 70, 602, 6078, 70402, 917830, 13253002, 209350350, 3584098770, 66012131222, 1300004931162, 27232369503902, 604103160535330, 14136908333006822, 347827448896896554, 8971450949011952494
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OFFSET
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0,2
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COMMENTS
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Original name: Sorting numbers.
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LINKS
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FORMULA
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E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=5.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=5. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (5*n/LambertW(5*n))^n * exp(n/LambertW(5*n) + (5*n/LambertW(5*n))^(1/5) - n - 6/5) / sqrt(1 + LambertW(5*n)). - Vaclav Kotesovec, Jul 10 2022
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MATHEMATICA
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u[0, j_]:=1; u[k_, j_]:=u[k, j]=Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 5], {n, 0, 12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 5; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n, k] * 5^k * BellB[k, 1/5] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)
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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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