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A033452 "STIRLING" transform of squares A000290. 5

%I

%S 0,1,5,22,99,471,2386,12867,73681,446620,2856457,19217243,135610448,

%T 1001159901,7714225057,61904585510,516347066551,4468588592739,

%U 40058673825258,371421499686007,3556976106133821

%N "STIRLING" transform of squares A000290.

%C 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

%C Convolved with powers of 2 = A058681: (1, 7, 36, 171, 813,...). Cf. triangle A180338. - _Gary W. Adamson_, Aug 28 2010

%F 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

%F a(n) = A005493(n) - A000110(n). - _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).)

%F a(n) = Bell(n+2) - 2*Bell(n+1). - _Vladeta Jovovic_, Jul 28 2003

%F G.f.: sum{k>=0, k^2*x^k/prod[l=1..k, 1-lx]}. - _Ralf Stephan_, Apr 18 2004

%F E.g.f.: exp( exp(x) - 1 + x) * (exp(x) - 1). - _Michael Somos_, Mar 28 2012

%F a(n) = A123158(n,3). - _Philippe Deléham_, Oct 06 2006

%F 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

%e G.f. = x + 5*x^2 + 22*x^3 + 99*x^4 + 471*x^5 + 2386*x^6 + 12867*x^7 + 73681*x^8 + ...

%p a:=n->(sum((j-1)*stirling2(n,j), j=2..n)): seq(a(n), n=2..21); # _Zerinvary Lajos_, Apr 18 2007

%t 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_ *)

%o (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 */

%Y Partial sums of A005494.

%Y Cf. A180338.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

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Last modified February 20 15:52 EST 2020. Contains 332078 sequences. (Running on oeis4.)