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%I #32 Jun 01 2019 09:34:31
%S 2,2,10,82,938,13778,247210,5240338,128149802,3551246162,109979486890,
%T 3764281873042,141104799067178,5749087305575378,252969604725106090,
%U 11955367835505775378,603967991604199335722,32479636694930586142802,1852497140997527094395050
%N Shifts left under "AIJ" (ordered, indistinct, labeled) transform.
%H Andrew Howroyd, <a href="/A032034/b032034.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = ((n-1)!*sum(k=1..n-1, binomial(n+k-1,n-1)*sum(j=1..k, (-1)^(j+n+1)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(j-l)!*2^(j-l)*(-1)^l*stirling2(n-l+j-1,j-l))/(n-l+j-1)!)))), n>1, a(1)=2. - _Vladimir Kruchinin_, Jan 24 2012
%F Let p(n,w) = w*Sum_{k=0..n-1} ((-1)^k*E2(n-1,k)*w^k)/(1+w)^(2*n-1),
%F E2 the second-order Eulerian numbers as defined by Knuth, then a(n) = p(n,-2). - _Peter Luschny_, Nov 10 2012
%F G.f.: 1 + 1/Q(0), where Q(k)= 1 + k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013
%F a(n) = 2 * A032188(n). - _Alois P. Heinz_, Jul 04 2018
%p with(combinat): A032034 := n -> add(eulerian2(n-1,k)*2^(k+1), k=0..n-1):
%p seq(A032034(n), n=1..17); # _Peter Luschny_, Nov 10 2012
%t Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n-1, k] (k+1) + Eulerian2[n-1, k-1] (2n-k-1)]];
%t a[n_] := Sum[Eulerian2[n-1, k] 2^(k+1), {k, 0, n-1}];
%t Array[a, 20] (* _Jean-François Alcover_, Jun 01 2019, after _Peter Luschny_ *)
%o (Maxima)
%o a(n):=if n=1 then 2 else ((n-1)!*sum(binomial(n+k-1,n-1)*sum((-1)^(j+n+1)*binomial(k,j)*sum((binomial(j,l)*(j-l)!*2^(j-l)*(-1)^l*stirling2(n-l+j-1,j-l))/(n-l+j-1)!,l,0,j),j,1,k),k,1,n-1)); /* _Vladimir Kruchinin_, Jan 24 2012 */
%o (Sage)
%o @CachedFunction
%o def eulerian2(n, k):
%o if k==0: return 1
%o elif k==n: return 0
%o return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
%o A032034 = lambda n: add(eulerian2(n-1,k)*2^(k+1) for k in (0..n-1))
%o [A032034(n) for n in (1..17)] # _Peter Luschny_, Nov 10 2012
%o (PARI) seq(n)={my(p=O(x)); for(i=1, n, p=intformal(1 + 1/(1-p))); Vec(serlaplace(p))} \\ _Andrew Howroyd_, Sep 19 2018
%Y Cf. A032188, A112487.
%K nonn
%O 1,1
%A _Christian G. Bower_
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