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A242817
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a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).
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23
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1, 1, 6, 57, 756, 12880, 268098, 6593839, 187104200, 6016681467, 216229931110, 8588688990640, 373625770888956, 17666550789597073, 902162954264563306, 49482106424507339565, 2901159958960121863952, 181069240855214001514460, 11985869691525854175222222
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051.
a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - Vaclav Kotesovec, May 23 2014
Conjecture: It appears that the equation a(x)*e^x = Sum_{n=0..oo} ( (n^x*x^n)/n! ) is true for every positive integer x. - Nicolas Nagel, Apr 20 2016 [This is just the special case k=x of the formula B(k,x) = e^(-x) * Sum_{n=0..oo} n^k*x^n/n!; see for example the World of Mathematics link. - Pontus von Brömssen, Dec 05 2020]
a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, May 31 2018
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1, (1+
add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
a:= n-> A(n$2):
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MATHEMATICA
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Table[BellB[n, n], {n, 0, 100}]
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PROG
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(Maxima) a(n):=stirling2(n, 0)+sum(stirling2(n, k)*n^k, k, 1, n);
makelist(a(n), n, 0, 30);
(PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*n^k); \\ Michel Marcus, Apr 20 2016
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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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