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

1,1

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

Christian G. Bower

Status

approved