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 A032034 Shifts left under "AIJ" (ordered, indistinct, labeled) transform. 3
 2, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802, 3551246162, 109979486890, 3764281873042, 141104799067178, 5749087305575378, 252969604725106090, 11955367835505775378, 603967991604199335722, 32479636694930586142802, 1852497140997527094395050 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 FORMULA 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 Let p(n,w) = w*Sum_{k=0..n-1} ((-1)^k*E2(n-1,k)*w^k)/(1+w)^(2*n-1), E2 the second-order Eulerian numbers as defined by Knuth, then a(n) = p(n,-2). - Peter Luschny, Nov 10 2012 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 a(n) = 2 * A032188(n). - Alois P. Heinz, Jul 04 2018 MAPLE with(combinat): A032034 := n -> add(eulerian2(n-1, k)*2^(k+1), k=0..n-1): seq(A032034(n), n=1..17); # Peter Luschny, Nov 10 2012 MATHEMATICA 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)]]; a[n_] := Sum[Eulerian2[n-1, k] 2^(k+1), {k, 0, n-1}]; Array[a, 20] (* Jean-François Alcover, Jun 01 2019, after Peter Luschny *) PROG (Maxima) 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 */ (Sage) @CachedFunction def eulerian2(n, k):     if k==0: return 1     elif k==n: return 0     return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1) A032034 = lambda n: add(eulerian2(n-1, k)*2^(k+1) for k in (0..n-1)) [A032034(n) for n in (1..17)]  # Peter Luschny, Nov 10 2012 (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 CROSSREFS Cf. A032188, A112487. Sequence in context: A326983 A232974 A181334 * A002250 A304642 A005613 Adjacent sequences:  A032031 A032032 A032033 * A032035 A032036 A032037 KEYWORD nonn AUTHOR STATUS approved

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Last modified May 18 03:03 EDT 2021. Contains 343994 sequences. (Running on oeis4.)