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A033452
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"STIRLING" transform of squares A000290.
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8
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0, 1, 5, 22, 99, 471, 2386, 12867, 73681, 446620, 2856457, 19217243, 135610448, 1001159901, 7714225057, 61904585510, 516347066551, 4468588592739, 40058673825258, 371421499686007, 3556976106133821, 35138574378189700, 357654857584636597, 3746672593640388775
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OFFSET
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0,3
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COMMENTS
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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, Oct 17 2002
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LINKS
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FORMULA
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Representation as an infinite series, in Maple notation: a(n)=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, Mar 21 2002
a(n) = A005493(n) - A000110(n+1). - Floor van Lamoen and _Christian Bower_, Oct 16 2002. (n^2 has e.g.f.: e^x * (x^2+x), a(n) thus has e.g.f: e^(e^x-1) * ( (e^x-1)^2 + (e^x-1) ) which simplifies to e^(e^x-1) * (e^2x - e^x). A005493 has e.g.f.: e^(e^x+2x-1), A000110 has e.g.f.: e^(e^x-1), A000110(n+1) has as e.g.f.: derivative of A000110 which is e^(e^x+x-1).) [corrected by Georg Fischer, Jun 17 2020]
G.f.: sum{k>=0, k^2*x^k/prod[l=1..k, 1-lx]}. - Ralf Stephan, Apr 18 2004
E.g.f.: exp( exp(x) - 1 + x) * (exp(x) - 1). - Michael Somos, Mar 28 2012
G.f.: G(0)/x -1/x, where G(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (2*x+x*k-1)*(3*x+x*k-1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 25 2014
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EXAMPLE
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G.f. = x + 5*x^2 + 22*x^3 + 99*x^4 + 471*x^5 + 2386*x^6 + 12867*x^7 + 73681*x^8 + ...
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MAPLE
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a := n -> add(Stirling2(n, j)*j^2, j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Apr 18 2007
# second Maple program:
b:= proc(n, m) option remember;
`if`(n=0, m^2, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 2 - 1/(1 - k*x)/(1 - x/(x - 1/g[k + 1])); gf = 1/x + 1/x^2 - g[0]/x^2; CoefficientList[ Series[gf, {x, 0, max}], x] (* Jean-François Alcover, Jan 24 2013, after Sergei N. Gladkovskii *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff( (exp(x + x * O(x^n)) - 1) * exp( exp(x + x * O(x^n)) - 1 + x), n))}; /* Michael Somos, Mar 28 2012 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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