A188062 Triangle of the value of Bell polynomials of the second kind B(n,m)(6,30,120,360,720,720) in row n, column m.

6, 30, 36, 120, 540, 216, 360, 5580, 6480, 1296, 720, 46800, 124200, 64800, 7776, 720, 331920, 1895400, 1976400, 583200, 46656, 0, 1995840, 24736320, 46947600, 25855200, 4898880, 279936, 0, 9979200, 284074560, 946527120, 876355200, 297198720, 39191040, 1679616, 0, 39916800, 2900620800

Offset: 1

Author

Vladimir Kruchinin, Mar 23 2011

Keywords

• nonn
• tabl

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.

Examples

			Table begins:
    6;
   30,    36;
  120,   540,    216;
  360,  5580,   6480,  1296;
  720, 46800, 124200, 64800, 7776;
		

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

Programs

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 *)

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);

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).