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A134055
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a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.
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15
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1, 1, 2, 8, 41, 252, 1782, 14121, 123244, 1169832, 11960978, 130742196, 1518514076, 18645970943, 241030821566, 3268214127548, 46338504902485, 685145875623056, 10538790233183702, 168282662416550040, 2784205185437851772, 47646587512911994120
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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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O.g.f.: Sum_{n>=0} (n*x)^n/(1-n*x)^n * exp(-n*x/(1-n*x)) / n!. - Paul D. Hanna, Nov 04 2012
a(n) mod 2 = A037011(n) for n >= 1.
a(n) mod 2 = 1 <=> n in { A048297 } or n = 0. (End)
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 252*x^5 + 1782*x^6 + 14121*x^7 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^2*x^2/(1-2*x)^2*exp(-2*x/(1-2*x))/2! + 3^3*x^3/(1-3*x)^3*exp(-3*x/(1-3*x))/3! + 4^4*x^4/(1-4*x)^4*exp(-4*x/(1-4*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41;
a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252;
a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782.
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MAPLE
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a:= proc(n) option remember; local b; b:=
proc(h, m) option remember; `if`(h=0,
binomial(n-1, m-1), m*b(h-1, m)+b(h-1, m+1) )
end; b(n, 0)
end:
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MATHEMATICA
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Flatten[{1, Table[Sum[Binomial[n-1, k-1] * StirlingS2[n, k], {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
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PROG
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(PARI) a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)))
(PARI) a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-k*x)^k*exp(-k*x/(1-k*x+x*O(x^n)))/k!), 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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An initial '1' was added and definition changed slightly by Paul D. Hanna, Nov 04 2012
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STATUS
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approved
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