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A304462 Coefficients of the compositionally inverted power series g:=f^{-1} of a formal power series f with the starting coefficients f_0=0 and f_1=1 expressed as polynomials in the coefficients f_2, f_3, ... of the given power series f(X) = X + f_2*X^2 + f_3*X^3 + ...

%I A304462 #32 Jun 24 2018 12:19:50
%S A304462 1,-1,-1,2,-1,5,-5,-1,6,3,-21,14,-1,7,7,-28,-28,84,-42,-1,8,8,-36,4,
%T A304462 -72,120,-12,180,-330,132,-1,9,9,-45,9,-90,165,-45,-45,495,-495,165,
%U A304462 -990,1287,-429
%N A304462 Coefficients of the compositionally inverted power series g:=f^{-1} of a formal power series f with the starting coefficients f_0=0 and f_1=1 expressed as polynomials in the coefficients f_2, f_3, ... of the given power series f(X) = X + f_2*X^2 + f_3*X^3 + ...
%C A304462 If g is taken as g(X) = X + g_2*X^2 + g_3*X^3 + ... then the compositions are (g circle f)(X) = g(f(X)) = X and (f circle g)(X) = f(g(X)) = X.
%C A304462 Lexicographically descending in the rows, i.e., f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4).
%C A304462 This is another version of A111785, where each row is sorted lexicographically ascending, i.e., f(1)^4 f(4)^2 (+4) < f(1)^3 f(2)^2 f(5) (-36).
%D A304462 Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 1953.
%F A304462 g(n) := f(1)^(-n) Sum_{j(2), j(3), ...} (-1)^{j(2) + j(3) + ...} ((n-1 + j(2) + j(3) + ...)!)/(n! j(2)! j(3)! ...) ((f(2))/(f(1))^j(2) ((f(3))/(f(1)))^j(3) ...
%F A304462 The sum is to be taken over all combinations of the exponents {j(2), j(3), j(4), ...} with j(2) + 2j(3) + 3j(4) + ... = n-1. See Morse, P. M. and Feshbach, H. pp. 411-413.
%e A304462 Matrix lexicographically descending in the rows:
%e A304462 for instance f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4)
%e A304462 1;
%e A304462 -1;
%e A304462 -1,2;
%e A304462 -1,5,-5;
%e A304462 -1,6,3,-21,14;
%e A304462 -1,7,7,-28,-28,84,-42;
%e A304462 -1,8,8,-36,4,-72,120,-12,180,-330,132;
%e A304462 -1,9,9,-45,9,-90,165,-45,-45,495,-495,165,-990,1287,-429;
%e A304462 -1,10,10,-55,10,-110,220,5,-110,-55,660,-715,-55,330,660,-2860,2002,55,-1430,5005,-5005,1430;
%o A304462 (MuPAD)
%o A304462 alfa:=["a","b","c","d","e","f","g","h","i","j","k"]:
%o A304462 byRow := proc(od, // original weighted degree
%o A304462 wd, // remaining weighted degree
%o A304462 il, // index of last indeterminate
%o A304462 jl, // exponent of last indeterminate
%o A304462 ni, // remaining number of indeterminates
%o A304462 lx) // lexicographic string
%o A304462 local j;
%o A304462 begin
%o A304462   if wd > 1 then
%o A304462     for j from min(wd,il) downto 2 do:
%o A304462       if j >= il then
%o A304462         j:=il: // stay at the latest indeterminate
%o A304462         byRow(od,wd-j+1,j,jl+1,ni-1,lx.alfa[j]):
%o A304462       else // advance to next indeterminate
%o A304462         byRow(od,wd-j+1,j,1   ,ni-1,lx.alfa[j]):
%o A304462       end_if:
%o A304462     end_for:
%o A304462   else // output the monomial
%o A304462     dd:=1: d0:="+": dc:=1:
%o A304462     for j from length(lx)-1 downto 0 do:
%o A304462       d1:=substring(lx,j):
%o A304462       if d1 <> d0 then
%o A304462         d0:=d1: dc:=1: dd:=-dd:
%o A304462       else // the indeterminate changes
%o A304462         dc:=dc+1: dd:=-dd*dc:
%o A304462       end_if:
%o A304462     end_for:
%o A304462     nn:=fact(2*od-ni-2)/fact(od): // rising factorial
%o A304462 // One row of A304462: coefficients of the lexicographically descending monomials:
%o A304462     print(nn/dd):
%o A304462 // One row of A304462: coefficients of the lexicographically descending monomials
%o A304462 // plus some representation of the monomials themselves:
%o A304462 //    for j from 1 to ni do:
%o A304462 //      lx:=lx."a":
%o A304462 //    end_for:
%o A304462 //    print(nn/dd,lx): // monomial lx
%o A304462   end_if:
%o A304462 end_proc:
%o A304462 // Output the 8th row:
%o A304462 n:=8:
%o A304462 byRow(n,n,n,0,n-1,"")
%Y A304462 Cf. A111785.
%K A304462 tabf,sign
%O A304462 0,4
%A A304462 _Herbert Eberle_, May 13 2018