A215269 - OEIS

%I #7 Aug 07 2012 11:33:56

%S 1,1,2,-3,2,-9,6,4,-36,75,-30,4,-60,255,-330,90,8,-180,1290,-3465,

%T 3150,-630,8,-252,2730,-12495,23940,-16380,2520,16,-672,10248,-71400,

%U 235305,-343980,185220,-22680,16,-864

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

%C 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.

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

%e 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.

%e 1;

%e 1;

%e 2,-3;

%e 2,-9,6;

%e 4,-36,75,-30;

%e 4,-60,255,-330,90;

%e 8,-180,1290,-3465,3150,-630;

%e 8,-252,2730,-12495,23940,-16380,2520;

%e 16,-672,10248,-71400,235305,-343980,185220,-22680;

%e 16,-864,17640,-173880,877905,-2226420,2593080,-1134000,113400;

%p H := proc(n,r,x)

%p     local e,d ;

%p     e := exp(-1/x^r) ;

%p     for d from 1 to n do

%p         e := diff(e,x) ;

%p     end do:

%p     x^((r+1)*n)*exp(1/x^r)*e ;

%p     expand(%) ;

%p end proc;

%p A215269 := proc(n,k)

%p     subs(x=sqrt(x),H(n,2,x))*2^(-floor(n/2)-(1-(-1)^n)/2) ;

%p     coeftayl( %,x=0,k) ;

%p end proc:

%p seq(seq( A215269(n,k),k=0..max(0,n-1)),n=0..12) ; # _R. J. Mathar_, Aug 07 2012

%Y Cf. A215216, A066667.

%K sign,tabf

%O 0,3

%A _Roman Witula_, Aug 07 2012