A138909 Expansion of e.g.f.: (1+x)/(1-x*exp(x)).

1, 2, 6, 33, 232, 2045, 21636, 267043, 3766848, 59776137, 1053986500, 20442543671, 432537117552, 9914571191005, 244742376434388, 6473030199939675, 182614314495736576, 5473825396372806545, 173728330742517310980

Offset

0,2

Comments

The n-th term of the n-th inverse binomial transform of this sequence equals n! for n>=0.

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f. (1+x)/(1-x*exp(x)) - Olivier Gérard, Sep 15 2016

O.g.f. satisfies: [x^n] A( x/(1+n*x) )/(1+n*x) = n! for n>=0.

E.g.f. satisfies: [x^n] A(x)*exp(-n*x) = 1 for n>=0.

a(n) = n! + Sum_{k=0..n-1} k!*C(n,k)*n*k^(n-k-1) for n>1 with a(0)=1.

Equivalent to the sum above by properties of the binomial triangle:

a(n) = A006153(n)+n*A006153(n-1).

a(n) = n! ( Sum_{k=0..n-1} ((n-1-k)^k + (n-k)^k)/k!) for n>1 with a(0)=1.

a(n) ~ n! / LambertW(1)^n. - Vaclav Kotesovec, Oct 30 2017

Example

If the successive inverse binomial transforms are placed in a table,
then we see that the diagonal consists of the factorials:
n=0:[(1),2,6,33,232,2045,21636,267043,3766848,59776137,1053986500,...];
n=1:[1,(1),3,20,129,1164,12265,151458,2136337,33901640,597761361,...];
n=2:[1,0,(2),13,64,693,6856,86175,1210896,19228825,339012304,...];
n=3:[1,-1,3,(6),25,482,3429,50908,678465,10937430,192150469,...];
n=4:[1,-2,6,-7,(24),381,844,36291,341728,6433865,107801436,...];
n=5:[1,-3,11,-32,97,(120),-839,37158,55953,4638052,54573025,...];
n=6:[1,-4,18,-75,304,-811,(720),40783,-262608,5542425,6069736,...];
n=7:[1,-5,27,-142,729,-3282,11941,(5040),-497279,9166130,...];
n=8:[1,-6,38,-239,1480,-8643,45844,-178557,(40320),12301705,...];
n=9:[1,-7,51,-372,2689,-18844,125289,-741974,3354513,(362880),...].

Mathematica

With[{nn=20}, CoefficientList[Series[(1+x)/(1-x*Exp[x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jun 09 2018 *)

Prog

(PARI) {a(n)=local(A=[1]); for(k=1, n, A=concat(A, 0); A[k+1]=k!-polcoeff(subst(Ser(A), x, x/(1+k*x+x*O(x^k)))/(1+k*x), k)); A[n+1]}
(PARI) {a(n)=n!+sum(k=0, n-1, k!*binomial(n, k)*n*k^(n-k-1))}

Crossrefs

Cf. A006153.

Cf. A138910 (inverse binomial transform); variants: A138911, A138737.

Sequence in context: A019028 A280769 A127114 * A138983 A121774 A209238

Adjacent sequences: A138906 A138907 A138908 * A138910 A138911 A138912

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2008, Apr 06 2008

Extensions

Name change and e.g.f. by Olivier Gérard, Sep 15 2016

Status

approved