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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.

%I A188062 #36 Feb 16 2025 08:33:14
%S A188062 6,30,36,120,540,216,360,5580,6480,1296,720,46800,124200,64800,7776,
%T A188062 720,331920,1895400,1976400,583200,46656,0,1995840,24736320,46947600,
%U A188062 25855200,4898880,279936,0,9979200,284074560,946527120,876355200,297198720,39191040,1679616,0,39916800,2900620800
%N 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.
%C A188062 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.
%H A188062 Ch. A. Charalambides, <a href="http://www.jstor.org/stable/25052047">On the generalized discrete distributions and the Bell polynomials</a>, Sankhya: Ind. J. Stat. B 39 (10) (1977) 36-44
%H A188062 F. T. Howard, <a href="http://dx.doi.org/10.1016/0012-365X(82)90136-4">A theorem relating potential and bell polynomials</a>, Discr. Math. 39 (2) (1982) 128-143.
%H A188062 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind </a>, arXiv:1104.5065 [math.CO], 2011.
%H A188062 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%F A188062 B(n,m) = n!/m!*sum_{k=0..m} binomial(m,k)*binomial(6*k,n)*(-1)^(m-k).
%F A188062 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).
%e A188062 Table begins:
%e A188062     6;
%e A188062    30,    36;
%e A188062   120,   540,    216;
%e A188062   360,  5580,   6480,  1296;
%e A188062   720, 46800, 124200, 64800, 7776;
%p A188062 # The function BellMatrix is defined in A264428.
%p A188062 # Adds (1,0,0,0, ..) as column 0.
%p A188062 BellMatrix(n -> `if`(n<6,[6,30,120,360,720,720][n+1],0), 9); # _Peter Luschny_, Jan 29 2016
%t A188062 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 *)
%t A188062 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A188062 rows = 12;
%t A188062 B = BellMatrix[Function[n, If[n<6, {6, 30, 120, 360, 720, 720}[[n+1]], 0]], rows];
%t A188062 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A188062 (Maxima)
%o A188062 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);
%K A188062 nonn,tabl
%O A188062 1,1
%A A188062 _Vladimir Kruchinin_, Mar 23 2011