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
KEYWORD
nonn
AUTHOR
STATUS
approved