A185106 Column 4 of A181783.

1, 7, 63, 709, 9709, 157971, 2993467, 64976353, 1593358809, 43632348319, 1321213523191, 43869502390077, 1585770335098693, 62013234471100459, 2609265444024424179, 117558236422872707161, 5647316731308685308337, 288166881285968665526583, 15566545814457889774570159, 887503412305357492886020789

Offset

0,2

Comments

A181783 is written as follows:

1, 1, 1, 1, 1, 1, 1, ...

1, 1, 2, 4, 7, 11, 16, ...

1, 1, 5, 21, 63, 151, 311, ...

1, 1, 16, 142, 709, 2521, ...

1, 1, 65, 1201, 9709, ...

A000522 and A053482 are respectively the columns number 2 and 3 of this array. Our sequence gives the column number 4 (the fifth).

Links

G. C. Greubel, Table of n, a(n) for n = 0..375

Formula

Recurrence relation: a(n)= 6*n*a(n-1) -11*n*(n-1)*a(n-2) +6*n*(n-1)*(n-2)*a(n-3) +1 (or following A053482 for a linear homogeneous recurrence) a(n)= (6n+1)*a(n-1) -(11n+6)*(n-1)*a(n-2) +(6n+11)*(n-1)*(n-2)*a(n-3) -6*(n-1)*(n-2)*(n-3)*a(n-4).

E.g.f: exp(z)/((1-z)*(1-2*z)*(1-3*z)), as explained in A181783.

With p=4, a(n)=a(n,p)=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n)

a(n) ~ n! * exp(1/3)*3^(n+2)/2. - Vaclav Kotesovec, Oct 02 2013

Maple

a(0, 1):=1:for p from 2 to 15 do for n from 0 to 20 do a(n, 0):=1 :a(n, p):=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)', 'k'=1..(p-1))', 'm'=0..n):od:seq(a(n, p), n=0..20):od;

Mathematica

CoefficientList[Series[E^x/((1-x)*(1-2x)*(1-3x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

Prog

(PARI) x='x+O('x^50); Vec(serlaplace(exp(x)/((1-x)*(1-2*x)*(1-3*x)))) \\ G. C. Greubel, Jun 22 2017

Crossrefs

Cf. A181783, A000522, A053482.

Sequence in context: A233743 A015684 A051579 * A275577 A049464 A229078

Adjacent sequences: A185103 A185104 A185105 * A185107 A185108 A185109

Keyword

nonn,easy

Author

Richard Choulet, Dec 26 2012

Status

approved