A032034 Shifts left under "AIJ" (ordered, indistinct, labeled) transform.

2, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802, 3551246162, 109979486890, 3764281873042, 141104799067178, 5749087305575378, 252969604725106090, 11955367835505775378, 603967991604199335722, 32479636694930586142802, 1852497140997527094395050

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A032188, A112487.

Programs

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 *)

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 */

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

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

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