OFFSET
0,3
COMMENTS
The modified Hermite-Bell polynomials for power -2 are defined by the formula H(n;-2;sqrt(x))*2^(-floor(n/2)-(1-(-1)^n)/2), where H(n;-2;x) denotes the n-th Hermite-Bell polynomial - see A215216 for the definition and details.
REFERENCES
R. Witula, E. Hetmaniok, D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)
EXAMPLE
If we set V(n;x):= H(n;-2;sqrt(x))*2^(-floor(n/2)-(1-(-1)^n)/2) then we obtain V(0;x)=V(1;x)=1, V(2;x)=2-3*x, V(3;x)=2-9*x+6*x^2, V(4;x)=4-36*x+75*x^2-30*x^3, V(5;x)=4-60*x+255*x^2-330*x^3+90*x^4.
1;
1;
2,-3;
2,-9,6;
4,-36,75,-30;
4,-60,255,-330,90;
8,-180,1290,-3465,3150,-630;
8,-252,2730,-12495,23940,-16380,2520;
16,-672,10248,-71400,235305,-343980,185220,-22680;
16,-864,17640,-173880,877905,-2226420,2593080,-1134000,113400;
MAPLE
H := proc(n, r, x)
local e, d ;
e := exp(-1/x^r) ;
for d from 1 to n do
e := diff(e, x) ;
end do:
x^((r+1)*n)*exp(1/x^r)*e ;
expand(%) ;
end proc;
A215269 := proc(n, k)
subs(x=sqrt(x), H(n, 2, x))*2^(-floor(n/2)-(1-(-1)^n)/2) ;
coeftayl( %, x=0, k) ;
end proc:
seq(seq( A215269(n, k), k=0..max(0, n-1)), n=0..12) ; # R. J. Mathar, Aug 07 2012
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Roman Witula, Aug 07 2012
STATUS
approved