OFFSET
1,1
COMMENTS
B(n,m)(6*x^5,30*x^4,120*x^3,360*x^2,720*x,720) = B(n,m)*x^(6*m-n) allows the computation of the Bell polynomials for a generalized set of arguments with a single parameter x.
LINKS
Ch. A. Charalambides, On the generalized discrete distributions and the Bell polynomials, Sankhya: Ind. J. Stat. B 39 (10) (1977) 36-44
F. T. Howard, A theorem relating potential and bell polynomials, Discr. Math. 39 (2) (1982) 128-143.
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind , arXiv:1104.5065 [math.CO], 2011.
Eric W. Weisstein, Bell Polynomial
FORMULA
B(n,m) = n!/m!*sum_{k=0..m} binomial(m,k)*binomial(6*k,n)*(-1)^(m-k).
B(n,m) = n!/m! *sum_{k=0..n-m} sum_{j=0..n} 3^j *binomial(j,n-3*k-3*m+2*j) *binomial(k+m,j) *binomial(m,k) *2^(m-k).
EXAMPLE
Table begins:
6;
30, 36;
120, 540, 216;
360, 5580, 6480, 1296;
720, 46800, 124200, 64800, 7776;
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n<6, [6, 30, 120, 360, 720, 720][n+1], 0), 9); # Peter Luschny, Jan 29 2016
MATHEMATICA
b[n_, m_] := n!/m!*Sum[ Sum[ 3^j*Binomial[j, n - 3*k - 3*m + 2*j]*Binomial[k + m, j], {j, 0, n}]*Binomial[m, k]*2^(m - k), {k, 0, n - m}]; Table[b[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013, translated from Maxima *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, If[n<6, {6, 30, 120, 360, 720, 720}[[n+1]], 0]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
PROG
(Maxima)
B(n, m):=n!/m!*sum(sum(3^j*binomial(j, n-3*k-3*m+2*j)*binomial(k+m, j), j, 0, n)*binomial(m, k)*2^(m-k), k, 0, n-m);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 23 2011
STATUS
approved