A007121 Expansion of e.g.f. ( (1+x)^x )^x.

1, 0, 0, 6, -12, 40, 180, -1512, 11760, -38880, 20160, 2106720, -22381920, 173197440, -703999296, -1737489600, 86030380800, -1149696737280, 11455162974720, -89560399541760, 636617260339200, -6318191386644480, 139398889956480000, -3797936822885990400

Offset: 0

Author

Simon Plouffe

Keywords

• sign

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..452

Crossrefs

Cf. A240989.

Programs

Maple

A007121 := proc(n)
        n!*coeftayl( (1+x)^(x^2),x=0,n) ;
end proc:
seq(A007121(n),n=0..40) ; # R. J. Mathar, Dec 15 2011

Mathematica

With[{nn=30},CoefficientList[Series[((1+x)^x)^x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 24 2014 *)

Maxima

a(n):=sum(stirling1(n-2*k, k)/(n-2*k)!, k, 0, n/3); /* Vladimir Kruchinin, Dec 13 2011 */

PARI

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 1)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=3, i, (-1)^j*j/(j-2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

Formula

a(n) = n!*Sum_{k=0..floor(n/3)} Stirling1(n-2*k,k)/(n-2*k)!. - Vladimir Kruchinin, Dec 13 2011

a(0) = 1; a(n) = -(n-1)! * Sum_{k=3..n} (-1)^k * k/(k-2) * a(n-k)/(n-k)!. - Seiichi Manyama, Jul 09 2022

Extensions

Signs added by R. J. Mathar, Vladimir Kruchinin, Dec 15 2011