OFFSET
0,3
COMMENTS
Compare with A134685 which uses a different order with fewer entries.
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.
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.
EXAMPLE
-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).
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Jul 14 2010
STATUS
approved