%I #32 Jun 24 2018 12:19:50

%S 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 -72,120,-12,180,-330,132,-1,9,9,-45,9,-90,165,-45,-45,495,-495,165,

%U -990,1287,-429

%N 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 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 Lexicographically descending in the rows, i.e., f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4).

%C 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 Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 1953.

%F 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 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 Matrix lexicographically descending in the rows:

%e for instance f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4)

%e 1;

%e -1;

%e -1,2;

%e -1,5,-5;

%e -1,6,3,-21,14;

%e -1,7,7,-28,-28,84,-42;

%e -1,8,8,-36,4,-72,120,-12,180,-330,132;

%e -1,9,9,-45,9,-90,165,-45,-45,495,-495,165,-990,1287,-429;

%e -1,10,10,-55,10,-110,220,5,-110,-55,660,-715,-55,330,660,-2860,2002,55,-1430,5005,-5005,1430;

%o (MuPAD)

%o alfa:=["a","b","c","d","e","f","g","h","i","j","k"]:

%o byRow := proc(od, // original weighted degree

%o wd, // remaining weighted degree

%o il, // index of last indeterminate

%o jl, // exponent of last indeterminate

%o ni, // remaining number of indeterminates

%o lx) // lexicographic string

%o local j;

%o begin

%o if wd > 1 then

%o for j from min(wd,il) downto 2 do:

%o if j >= il then

%o j:=il: // stay at the latest indeterminate

%o byRow(od,wd-j+1,j,jl+1,ni-1,lx.alfa[j]):

%o else // advance to next indeterminate

%o byRow(od,wd-j+1,j,1 ,ni-1,lx.alfa[j]):

%o end_if:

%o end_for:

%o else // output the monomial

%o dd:=1: d0:="+": dc:=1:

%o for j from length(lx)-1 downto 0 do:

%o d1:=substring(lx,j):

%o if d1 <> d0 then

%o d0:=d1: dc:=1: dd:=-dd:

%o else // the indeterminate changes

%o dc:=dc+1: dd:=-dd*dc:

%o end_if:

%o end_for:

%o nn:=fact(2*od-ni-2)/fact(od): // rising factorial

%o // One row of A304462: coefficients of the lexicographically descending monomials:

%o print(nn/dd):

%o // One row of A304462: coefficients of the lexicographically descending monomials

%o // plus some representation of the monomials themselves:

%o // for j from 1 to ni do:

%o // lx:=lx."a":

%o // end_for:

%o // print(nn/dd,lx): // monomial lx

%o end_if:

%o end_proc:

%o // Output the 8th row:

%o n:=8:

%o byRow(n,n,n,0,n-1,"")

%Y Cf. A111785.

%K tabf,sign

%O 0,4

%A _Herbert Eberle_, May 13 2018