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A033452 "STIRLING" transform of squares A000290.

%I A033452 #78 May 04 2025 01:54:56
%S A033452 0,1,5,22,99,471,2386,12867,73681,446620,2856457,19217243,135610448,
%T A033452 1001159901,7714225057,61904585510,516347066551,4468588592739,
%U A033452 40058673825258,371421499686007,3556976106133821,35138574378189700,357654857584636597,3746672593640388775
%N A033452 "STIRLING" transform of squares A000290.
%C A033452 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 A033452 Convolved with powers of 2 = A058681: (1, 7, 36, 171, 813, ...). Cf. triangle A180338. - _Gary W. Adamson_, Aug 28 2010
%H A033452 Alois P. Heinz, <a href="/A033452/b033452.txt">Table of n, a(n) for n = 0..574</a>
%F A033452 Representation as an infinite series: a(n) = (Sum_{k>=1} k^n*k*(k-2)/k!)/exp(1), n >= 1. This is a Dobinski-type summation formula. - _Karol A. Penson_, Mar 21 2002
%F A033452 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]
%F A033452 a(n) = Bell(n+2) - 2*Bell(n+1). - _Vladeta Jovovic_, Jul 28 2003
%F A033452 G.f.: sum{k>=0, k^2*x^k/prod[l=1..k, 1-lx]}. - _Ralf Stephan_, Apr 18 2004
%F A033452 E.g.f.: exp( exp(x) - 1 + x) * (exp(x) - 1). - _Michael Somos_, Mar 28 2012
%F A033452 a(n) = A123158(n,3). - _Philippe Deléham_, Oct 06 2006
%F A033452 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 A033452 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 A033452 a := n -> add(Stirling2(n, j)*j^2, j=0..n): seq(a(n), n=0..20); # _Zerinvary Lajos_, Apr 18 2007
%p A033452 # second Maple program:
%p A033452 b:= proc(n, m) option remember;
%p A033452      `if`(n=0, m^2, m*b(n-1, m)+b(n-1, m+1))
%p A033452     end:
%p A033452 a:= n-> b(n, 0):
%p A033452 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 04 2021
%t A033452 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 A033452 (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 A033452 Partial sums of A005494.
%Y A033452 Cf. A180338.
%K A033452 nonn
%O A033452 0,3
%A A033452 _N. J. A. Sloane_