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

%I A215269 #7 Aug 07 2012 11:33:56
%S A215269 1,1,2,-3,2,-9,6,4,-36,75,-30,4,-60,255,-330,90,8,-180,1290,-3465,
%T A215269 3150,-630,8,-252,2730,-12495,23940,-16380,2520,16,-672,10248,-71400,
%U A215269 235305,-343980,185220,-22680,16,-864
%N A215269 Coefficient triangle of the modified Hermite-Bell polynomials for power -2.
%C A215269 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 A215269 R. Witula, E. Hetmaniok, D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)
%e A215269 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 A215269 1;
%e A215269 1;
%e A215269 2,-3;
%e A215269 2,-9,6;
%e A215269 4,-36,75,-30;
%e A215269 4,-60,255,-330,90;
%e A215269 8,-180,1290,-3465,3150,-630;
%e A215269 8,-252,2730,-12495,23940,-16380,2520;
%e A215269 16,-672,10248,-71400,235305,-343980,185220,-22680;
%e A215269 16,-864,17640,-173880,877905,-2226420,2593080,-1134000,113400;
%p A215269 H := proc(n,r,x)
%p A215269     local e,d ;
%p A215269     e := exp(-1/x^r) ;
%p A215269     for d from 1 to n do
%p A215269         e := diff(e,x) ;
%p A215269     end do:
%p A215269     x^((r+1)*n)*exp(1/x^r)*e ;
%p A215269     expand(%) ;
%p A215269 end proc;
%p A215269 A215269 := proc(n,k)
%p A215269     subs(x=sqrt(x),H(n,2,x))*2^(-floor(n/2)-(1-(-1)^n)/2) ;
%p A215269     coeftayl( %,x=0,k) ;
%p A215269 end proc:
%p A215269 seq(seq( A215269(n,k),k=0..max(0,n-1)),n=0..12) ; # _R. J. Mathar_, Aug 07 2012
%Y A215269 Cf. A215216, A066667.
%K A215269 sign,tabf
%O A215269 0,3
%A A215269 _Roman Witula_, Aug 07 2012