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A134055 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.

%I A134055 #26 Jun 24 2023 22:06:37
%S A134055 1,1,2,8,41,252,1782,14121,123244,1169832,11960978,130742196,
%T A134055 1518514076,18645970943,241030821566,3268214127548,46338504902485,
%U A134055 685145875623056,10538790233183702,168282662416550040,2784205185437851772,47646587512911994120
%N A134055 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.
%H A134055 Alois P. Heinz, <a href="/A134055/b134055.txt">Table of n, a(n) for n = 0..518</a>
%F A134055 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
%F A134055 From _Alois P. Heinz_, Jun 24 2023: (Start)
%F A134055 a(n) mod 2 = A037011(n) for n >= 1.
%F A134055 a(n) mod 2 = 1 <=> n in { A048297 } or n = 0. (End)
%e A134055 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 +...
%e A134055 where
%e A134055 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! +...
%e A134055 simplifies to a power series in x with integer coefficients.
%e A134055 Illustrate the definition of the terms by:
%e A134055 a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41;
%e A134055 a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252;
%e A134055 a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782.
%p A134055 a:= proc(n) option remember; local b; b:=
%p A134055       proc(h, m) option remember; `if`(h=0,
%p A134055         binomial(n-1, m-1), m*b(h-1, m)+b(h-1, m+1) )
%p A134055       end; b(n, 0)
%p A134055     end:
%p A134055 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jun 24 2023
%t A134055 Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n,k],{k,1,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Aug 11 2014 *)
%o A134055 (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)))
%o A134055 (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)
%o A134055 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 04 2012
%Y A134055 Cf. A037011, A048297, A048993, A122455, A218667, A218670, A245059, A245060.
%K A134055 nonn
%O A134055 0,3
%A A134055 _Paul D. Hanna_, Oct 08 2007
%E A134055 An initial '1' was added and definition changed slightly by _Paul D. Hanna_, Nov 04 2012