A045406 A diagonal of A008296.

1, 3, -1, 0, 4, -28, 188, -1368, 11016, -98208, 964512, -10370880, 121337280, -1535880960, 20924455680, -305396421120, 4755302899200, -78700195123200, 1379748896870400, -25546854999859200, 498194992408780800, -10207190048993280000, 219216795045212160000

Offset

2,2

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,2).

Links

Robert Israel, Table of n, a(n) for n = 2..414

Formula

a(n) = A008296(n,2).

E.g.f.: ((1+x)*log(1+x))^2/2. - Vladeta Jovovic, Feb 20 2003

a(n) = sum(i=1, n-1, i^2*Stirling1(n-1, i)). - Benoit Cloitre, Oct 23 2004

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = f(n,2,-2), for n>=2. - Milan Janjic, Dec 21 2008

a(n) = (-1)^(n)*(2*H(n-3)-3)*(n-3)! for n >= 3, where H(n) = Sum(j=1..n, 1/j) is the n-th harmonic number. - Gary Detlefs, Feb 13 2010

a(n) = 2*A081048(n-3)-3*(-1)^(n)*(n-3)! for n >= 3. - Robert Israel, Jun 28 2015

Sum_{k=1..n} a(k+1) * Stirling2(n,k) = n^2. - Vaclav Kotesovec, Sep 03 2018

Conjecture: D-finite with recurrence a(n) +(2*n-7)*a(n-1) +(n-4)^2*a(n-2)=0. - R. J. Mathar, Sep 15 2021

Maple

with(combinat): for n from 2 to 40 do for k from 2 to 2 do printf(`%d, `, sum(binomial(l, k)*k^(l-k)*stirling1(n, l), l=k..n)) od: od:
# Alternative:
A081048:= gfun:-rectoproc({a(0)=0, a(1)=1, a(n)=(1-2*n)*a(n-1) -(n-1)^2*a(n-2)}, a(n), remember):
1, seq(2*A081048(n-3)-3*(-1)^(n)*(n-3)!, n=3..50); # Robert Israel, Jun 29 2015

Mathematica

With[{nn=30}, CoefficientList[Series[((1+x)Log[1+x])^2/2, {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Jun 04 2019 *)

Crossrefs

Cf. A081048.

Sequence in context: A207327 A319083 A332099 * A143468 A133728 A127627

Adjacent sequences: A045403 A045404 A045405 * A045407 A045408 A045409

Keyword

sign,easy

Author

N. J. A. Sloane, Jan 26 2001

Extensions

More terms from James A. Sellers, Jan 26 2001

Gary Detlefs comment changed to a formula by Robert Israel, Jun 28 2015

Status

approved