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%I #10 Nov 05 2016 08:09:43
%S 1,-1,1,2,-3,1,-7,12,-6,1,37,-67,39,-10,1,-266,495,-310,95,-15,1,2431,
%T -4596,3000,-1010,195,-21,1,-27007,51583,-34566,12320,-2660,357,-28,1,
%U 353522,-680037,463981,-171766,39795,-6062,602,-36,1,-5329837,10306152,-7124454,2709525,-658791,108927,-12432,954
%N Matrix inverse of A100862.
%C Column 0 is signed A001515 (Bessel polynomial). Column 1 is A107103. Row sums are zeros for n>0. Absolute row sums form A107104, which equals 2*A043301(n-1) for n>0.
%C The row polynomials p_n(x) of this entry are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials of A001497, e,g, (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - _Tom Copeland_, Oct 10 2016
%F E.g.f.: exp((1-y)*(1-sqrt(1+2*x))). [_Vladeta Jovovic_, Dec 13 2008]
%e Triangle begins:
%e 1;
%e -1,1;
%e 2,-3,1;
%e -7,12,-6,1;
%e 37,-67,39,-10,1;
%e -266,495,-310,95,-15,1;
%e 2431,-4596,3000,-1010,195,-21,1;
%e -27007,51583,-34566,12320,-2660,357,-28,1; ...
%e and is the matrix inverse of A100862:
%e 1;
%e 1,1;
%e 1,3,1;
%e 1,6,6,1;
%e 1,10,21,10,1;
%e 1,15,55,55,15,1; ...
%o (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^n));(matrix(n+1,n+1,m,j,if(m>=j, (m-1)!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y),m-1,x),j-1,y)))^-1)[n+1,k+1]}
%Y Cf. A100862, A001515, A043301, A107103, A107104.
%Y Cf. A001497.
%K sign,tabl
%O 0,4
%A _Paul D. Hanna_, May 21 2005
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