A187082 - OEIS

%I #19 Jul 15 2019 17:36:20

%S 3,6,9,6,54,27,0,180,324,81,0,360,2160,1620,243,0,360,9720,17820,7290,

%T 729,0,0,30240,136080,119070,30618,2187,0,0,60480,771120,1360800,

%U 694008,122472,6561,0,0,60480,3265920,11838960,11022480,3674160,472392,19683

%N Bell polynomial B(n,k){3,6,6,0,...,0}

%C B(n,k){3*x^2,6*x,6,0,...,0)=n!/k!*x^(3*k-n)*sum(3^j*binomial(j,n-3*k+2*j)*binomial(k,j),j,0,n).

%C The Bell transform of the sequence "a(n) = 3,6,6,0,0,0, ..." without column 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065, 2011.

%H Jin Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.html">Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

%F B(n,k) = n!/k!*sum(j=0..n, 3^j*binomial(j,n-3*k+2*j)*binomial(k,j));

%e [3],

%e [6,9],

%e [6,54,27],

%e [0,180,324,81],

%e [0,360,2160,1620,243],

%e [0,360,9720,17820,7290,729],

%e [0,0,30240,136080,119070,30618,2187],

%e [0,0,60480,771120,1360800,694008,122472,6561],

%e [0,0,60480,3265920,11838960,11022480,3674160,472392,19683]

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> `if`(n<3,[3,6,6][n+1],0), 10); # _Peter Luschny_, Jan 29 2016

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, If[n < 3, {3, 6, 6}[[n + 1]], 0]], rows = 12];

%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%K nonn,tabl

%O 1,1

%A _Vladimir Kruchinin_, Mar 03 2011