A176740 Inversion of e.g.f. formal power series. Partition array in Abramowitz-Stegun (A-St) order.

-1, -1, 3, -1, 10, -15, -1, 15, 10, -105, 105, -1, 21, 35, -210, -280, 1260, -945, -1, 28, 56, 35, -378, -1260, -280, 3150, 6300, -17325, 10395, -1, 36, 84, 126, -630, -2520, -1575, -2100, 6930, 34650, 15400, -51975, -138600, 270270, -135135, -1, 45, 120, 210, 126, -990, -4620, -6930, -4620, -5775

Offset: 0

Author

Wolfdieter Lang, Jul 14 2010

Keywords

• sign
• easy
• tabf

Comments

Compare with A134685 which uses a different order with fewer entries.

For the inversion (aka reversion) of o.g.f. formal power series see A111785, and also A133437.

The sequence of row lengths of this array is p(n)=A000041(n) (number of partitions of n).

The unsigned triangle, with entries for like parts number m summed, is A134991 (2-associated Stirling numers of the second kind).

The row sums are A133942(n) = ((-1)^n) * n!, and the row sums of the unsigned array give A000311(n+1) (Schroeder's fourth problem). These sums coincide with those of the triangle A134991.

The signed a(n,k) numbers, k=1,...,p(n)=A000041(n), derive from the multinomial M_3 numbers A036040 (see also the W. Lang link there), namely, if the k-th partition of n in A-St order has exponents (enk[1],...,enk[n]) then a(n,k) = ((-1)^m)*M3(n+m, (ehatnk[1],...,ehatnk[n+m])) with m the number of parts, i.e., m:=Sum_{j=1..n} enk[j], and M3(n+m, (ehatnk[1],...,ehatnk[n+m])):=(n+m)!/(Product_{j=1..n+m} j!^ehatnk[j]*ehatnk[j]!), where the n+m exponents ehatnk are ehatnk[1]:=0, (ehatnk[2],...,ehatnk[n+1]) := (enk[1],...,enk[n]), and (ehatnk[n+1],...,ehatnk[n+m]):=(0,...,0) (i.e., m-1 zeros).

The compositional inverse of the formal power series of the e.g.f. type g(x) = Sum_{j>=1} g[j]*(x^j)/j! is f = g^[-1] with f(y) = Sum_{n>=1} f[n]*(y^n)/n!, and f[n] = fhat[n]/g[1]^(2*n-1) with fhat[1]=1 (f[1] = 1/g[1]) and f[n+1] = Sum_{k=1..p(n)} a(n,k)*g(n,k), n >= 1, where p(n) = A000041(n) (number of partitions of n), and g(n,k) is the monomial in coefficients of g(x) corresponding to the k-th partition of 2*n with n parts in A-St order. For details and a remark on the Faa di Bruno Hopf algebra see the W. Lang link.

Examples

			  -1;
  -1,  3;
  -1, 10, -15;
  -1, 15,  10, -105,  105;
  -1, 21,  35, -210, -280, 1260, -945;
...
a(4,4): 4th partition of 4 has exponents (2,1,0,0) with m=3, and the derived exponents ehatm are (0,2,1,0,0,0,0) with one leading and 2 extra trailing zeros. (4+3)!/(2!^2*2!*3!^1*1!) = 105, hence a(4,4) = ((-1)^3)*105 = -105.
fhat[4] = -1*g[1]^2*g[4] +10*g[1]*g[2]*g[3] - 15*g[2]^3 (n=3: 3 parts partitions of 6 for the g-monomials in A-St order).
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831-2.
• R. Aldrovandi, Special Matrices of Mathematical Physics, World Scientific, 2001, p. 175, eq. (13.84).
• Ch. A. Charalambides, Enumerative Combinatorics, Chapman &Hall/CRC, 2002, p. 437, eq. (11.43) with p. 428. eq. (11.29).

Links

• W. P. Johnson, Combinatorics of Higher Derivatives of Inverses, Amer. Math. Monthly 109 (3), (2002), 273-277
• Wolfdieter Lang, E.g.f. Lagrange inversion partition array.

Formula

See the fhat[n] formula explained above, and the W. Lang link for more details.