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 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. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified May 27 03:13 EDT 2022. Contains 354093 sequences. (Running on oeis4.)