0,3

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.

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).

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.

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

-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).

Sequence in context: A146154 A068438 A064060 * A370258 A134991 A212930

Adjacent sequences: A176737 A176738 A176739 * A176741 A176742 A176743

sign,easy,tabf

Wolfdieter Lang, Jul 14 2010

approved