A187082 Bell polynomial B(n,k){3,6,6,0,...,0}

3, 6, 9, 6, 54, 27, 0, 180, 324, 81, 0, 360, 2160, 1620, 243, 0, 360, 9720, 17820, 7290, 729, 0, 0, 30240, 136080, 119070, 30618, 2187, 0, 0, 60480, 771120, 1360800, 694008, 122472, 6561, 0, 0, 60480, 3265920, 11838960, 11022480, 3674160, 472392, 19683

Offset

1,1

Comments

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).

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

Links

Table of n, a(n) for n=1..45.

Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065, 2011.

Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

Formula

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

Example

[3],
[6,9],
[6,54,27],
[0,180,324,81],
[0,360,2160,1620,243],
[0,360,9720,17820,7290,729],
[0,0,30240,136080,119070,30618,2187],
[0,0,60480,771120,1360800,694008,122472,6561],
[0,0,60480,3265920,11838960,11022480,3674160,472392,19683]

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n<3, [3, 6, 6][n+1], 0), 10); # Peter Luschny, Jan 29 2016

Mathematica

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[n < 3, {3, 6, 6}[[n + 1]], 0]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Crossrefs

Sequence in context: A049341 A321943 A351189 * A377030 A137991 A021077

Adjacent sequences: A187079 A187080 A187081 * A187083 A187084 A187085

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 03 2011

Status

approved