1,5

T[n,m]=count of set partitions of n with block lengths given by the m-th partition of n.

Contribution from Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008): (Start)

These are also the coefficients occuring in complete Bell polynomials, Faa di Bruno's formula (in it's simplest form), and computation of moments from cumulants.

Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (see e.g. [Coffey], and program below)

The complete Bell polynomial of the first n primes gives A007446.

(End)

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_3".

T. D. Noe, Rows n=1..25 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

W. Lang: Array and polynomials.

Wikipedia, Bell polynomials [From Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008]

Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications [From Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008]

E.g.f. A(t)= exp(sum(x[k]*(t^k)/k!,k=1..infinity)).

T(n,m) is the coefficient of ((t^n)/n!)* x[1]^e(m,1)*x[2]^e(m,2)*...*x[n]^e(m,n) in A(t). Here the m-th partition of n, counted in Abramowitz-Stegun(A-St) order, is [1^e(m,1), 2^e(m,2), ..., n^e(m,n)] with e(m,j)>=0, and if e(m, j)=0 then j^0 is not recorded.

a(n, m)= n!/product((j!^e(m,j))*e(m,j)!,j=1..n ), with [1^e(m,1),2^e(m,2), ...,n^e(m, n)] the m-th partition of n in the mentioned A-St order.

With the notation in the Lang reference, x(1) treated as a variable, and D the derivative w.r.t. x(1), a raising operator for the polynomial S(n,x(1)) = P3_n(x[1],...,x[n]) is R = sum(n=0,1,...) x(n+1) D^n / n! ; i.e., R S(n,x(1)) = S(n+1,x(1)). The lowering operator is D ; i.e., D S(n,x(1)) = n S(n-1,x(1)). The sequence of polynomials is an Appell sequence, so [S(.,x(1))+y]^n = S(n,x(1)+y). For x(j) = (-1)^(j-1) (j-1)! for j>1, S(n,x(1)) = [x(1)-1]^n + n [x(1)-1]^(n-1). [From Tom Copeland (tcjpn(AT)msn.com), Aug 01 2008]

1; 1,1; 1,3,1; 1,4,3,6,1; ...

<<DiscreteMath`Combinatorica`; runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[temp=Map[Reverse, Sort@ (Sort/@ Partitions[w]), {1}]; Apply[Multinomial, temp, {1}]/Apply[Times, (runs/@ temp)!, {1}], {w, 6}]

Contribution from Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008: (Start)

(Other) completeBellMatrix := proc(x, n)

// x - vector x[1]...x[m], m>=n

local i, j, M;

begin

M:=matrix(n, n): // zero-initialized

for i from 1 to n-1 do

M[i, i+1]:=-1:

end_for:

for i from 1 to n do

for j from 1 to i do

M[i, j] := binomial(i-1, j-1)*x[i-j+1]:

end_for:

end_for:

return (M):

end_proc:

completeBellPoly := proc(x, n)

begin

return (linalg::det(completeBellMatrix(x, n))):

end_proc:

for i from 1 to 10 do print(i, completeBellPoly(x, i)): end_for:

(End)

See A080575 for another version. Cf. A036036-A036039.

Row sums are the Bell numbers A000110.

Adjacent sequences: A036037 A036038 A036039 this_sequence A036041 A036042 A036043

Sequence in context: A125181 A049999 A126015 this_sequence A080575 A077228 A049687

Cf. A000040, A007446 [From Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 05 2008]

nonn,easy,nice,tabf,new

njas

More terms from David W. Wilson (davidwwilson(AT)comcast.net).

Additional comments from Wouter Meeussen, Mar 23, 2003