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A033452
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"STIRLING" transform of squares A000290.
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4
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1, 5, 22, 99, 471, 2386, 12867, 73681, 446620, 2856457, 19217243, 135610448, 1001159901, 7714225057, 61904585510, 516347066551, 4468588592739, 40058673825258, 371421499686007, 3556976106133821
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| If an integer N is squarefree and has n+2 distinct prime factors then a(n) is the number of product signs needed to write the factorizations of N, so a(n)=A076277(N) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 17 2002.
Convolved with powers of 2 = A058681: (1, 7, 36, 171, 813,...). Cf. triangle A180338. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2010]
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FORMULA
| Representation as an infinite series, in Maple notation : a(n-1)=sum(k^n*k*(k-2)/k!, k=1..infinity)/exp(1), n=1, 2... . This is a Dobinski-type summation formula. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Mar 21, 2002
a(n) = A005493(n)-A000110(n+1). - Floor en Lyanne van Lamoen (fvlamoen(AT)hotmail.com) and Chritain Bower, Oct 16 2002. (n^2 has egf e^x * (x^2+x), a(n) thus has egf e^(e^x-1) * ( (e^x-1)^2 + (e^x-1) ) which simplifies to e^(e^x-1) * (e^2x - e^x). A005493 has egf e^(e^x+2x-1), A000110 has egf e^(e^x-1), A000110(n+1) has as egf derivative of A000110 which is e^(e^x+x-1).)
a(n) = Bell(n+2)-2*Bell(n+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 28 2003
G.f.: sum{k>=0, k^2*x^k/prod[l=1..k, 1-lx]}. - R. Stephan, Apr 18 2004
a(n) = A123158(n,3) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 06 2006
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MAPLE
| a:=n->(sum((j-1)*stirling2(n, j), j=2..n)): seq(a(n), n=2..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 18 2007
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CROSSREFS
| Partial sums of A005494.
Cf. A180338 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2010]
Sequence in context: A129164 A123347 A087439 * A179602 A048251 A017971
Adjacent sequences: A033449 A033450 A033451 * A033453 A033454 A033455
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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