A215269 Coefficient triangle of the modified Hermite-Bell polynomials for power -2.

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

Offset: 0

Author

Roman Witula, Aug 07 2012

Keywords

• sign
• tabf

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.

Examples

			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;
		

References

• R. Witula, E. Hetmaniok, D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

Crossrefs

Cf. A215216, A066667.

Programs

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