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A036082
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E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=12.
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0
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1, 2, 17, 231, 3724, 68819, 1464781, 35645040, 973624491, 29313919207, 960689482494, 33997330377817, 1291521482389621, 52395164853506674, 2259005857941805253, 103064324686839195035, 4957382457319437575820, 250592665906288206715951, 13275467282249493427541201
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OFFSET
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0,2
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COMMENTS
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In general, for p>=2, a(n) ~ c * (p*n/LambertW(p*n))^n * exp(n/LambertW(p*n) + (p*n/LambertW(p*n))^(1/p) - n - 1 - 1/p) / sqrt(1 + LambertW(p*n)), where c = 1 for p>=3 and c = exp(-1/4) for p=2. - Vaclav Kotesovec, Jul 10 2022
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REFERENCES
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T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
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LINKS
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FORMULA
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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=12. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (12*n/LambertW(12*n))^n * exp(n/LambertW(12*n) + (12*n/LambertW(12*n))^(1/12) - n - 13/12) / sqrt(1 + LambertW(12*n)). - Vaclav Kotesovec, Jul 10 2022
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MATHEMATICA
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mx = 16; p = 12; 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] * 12^k * BellB[k, 1/12] * 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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