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

1,1

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

Table of n, a(n) for n=1..39.

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

Sequence in context: A175497 A161812 A282944 * A056153 A062515 A316532

Adjacent sequences: A188059 A188060 A188061 * A188063 A188064 A188065

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 23 2011

Status

approved