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 + ...

1, -1, -1, 2, -1, 5, -5, -1, 6, 3, -21, 14, -1, 7, 7, -28, -28, 84, -42, -1, 8, 8, -36, 4, -72, 120, -12, 180, -330, 132, -1, 9, 9, -45, 9, -90, 165, -45, -45, 495, -495, 165, -990, 1287, -429

Offset: 0

Author

Herbert Eberle, May 13 2018

Keywords

• tabf
• sign

Comments

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.

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

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).

Examples

			Matrix lexicographically descending in the rows:
for instance f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4)
1;
-1;
-1,2;
-1,5,-5;
-1,6,3,-21,14;
-1,7,7,-28,-28,84,-42;
-1,8,8,-36,4,-72,120,-12,180,-330,132;
-1,9,9,-45,9,-90,165,-45,-45,495,-495,165,-990,1287,-429;
-1,10,10,-55,10,-110,220,5,-110,-55,660,-715,-55,330,660,-2860,2002,55,-1430,5005,-5005,1430;
		

References

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

Crossrefs

Cf. A111785.

Programs

MuPAD

alfa:=["a","b","c","d","e","f","g","h","i","j","k"]:
byRow := proc(od, // original weighted degree
wd, // remaining weighted degree
il, // index of last indeterminate
jl, // exponent of last indeterminate
ni, // remaining number of indeterminates
lx) // lexicographic string
local j;
begin
  if wd > 1 then
    for j from min(wd,il) downto 2 do:
      if j >= il then
        j:=il: // stay at the latest indeterminate
        byRow(od,wd-j+1,j,jl+1,ni-1,lx.alfa[j]):
      else // advance to next indeterminate
        byRow(od,wd-j+1,j,1   ,ni-1,lx.alfa[j]):
      end_if:
    end_for:
  else // output the monomial
    dd:=1: d0:="+": dc:=1:
    for j from length(lx)-1 downto 0 do:
      d1:=substring(lx,j):
      if d1 <> d0 then
        d0:=d1: dc:=1: dd:=-dd:
      else // the indeterminate changes
        dc:=dc+1: dd:=-dd*dc:
      end_if:
    end_for:
    nn:=fact(2*od-ni-2)/fact(od): // rising factorial
// One row of A304462: coefficients of the lexicographically descending monomials:
    print(nn/dd):
// One row of A304462: coefficients of the lexicographically descending monomials
// plus some representation of the monomials themselves:
//    for j from 1 to ni do:
//      lx:=lx."a":
//    end_for:
//    print(nn/dd,lx): // monomial lx
  end_if:
end_proc:
// Output the 8th row:
n:=8:
byRow(n,n,n,0,n-1,"")

Formula

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) ...

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.