A188066 Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7, 42, 210, 840, 2520, 5040, 5040).

7, 42, 49, 210, 882, 343, 840, 11172, 12348, 2401, 2520, 117600, 288120, 144060, 16807, 5040, 1076040, 5433120, 5330220, 1512630, 117649, 5040, 8643600, 89029080, 155296680, 81177810, 14823774, 823543, 0, 60540480, 1306912320, 3884433840, 3360055440, 1087076760, 138355224, 5764801

Offset

1,1

Comments

From the explicit write-up of the Bell polynomials we have B(n,k)(7*x^6, 42*x^5, 210*x^4, 840*x^3, 2520*x^2, 5040*x, 5040) = B(n,k)(7, 42, ..., 5040)*x^(7*k-n) for a more general set of arguments.

Links

Table of n, a(n) for n=1..36.

M. Abbas and S. Bouroubi, On new identities for Bell's polynomials, Disc. Math. 293 (1-3) (2005), 5-10.

Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.

John Riordan, Derivatives of composite functions, Bull. Am. Math. Soc. 52 (1946), 664-667.

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

B(n,k) = (n!/k!)*Sum_{j=0..k} binomial(k,j)*binomial(7*j,n)*(-1)^(k-j).

Example

Triangle begins
     7;
    42,      49;
   210,     882,     343;
   840,   11172,   12348,    2401;
  2520,  117600,  288120,  144060,   16807;
  5040, 1076040, 5433120, 5330220, 1512630, 117649;
  ...

Maple

A188066 := proc(n, k) n!/k!*add( binomial(k, j)*binomial(7*j, n)*(-1)^(k-j), j=0..k) ; end proc:
seq(seq(A188066(n, k), k=1..n), n=1..5) ; # R. J. Mathar, Apr 08 2011
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n<7, [7, 42, 210, 840, 2520, 5040, 5040][n+1], 0), 9); # Peter Luschny, Jan 29 2016

Mathematica

b[n_, k_] := n!/k!*Sum[ Binomial[k, j]*Binomial[7*j, n]*(-1)^(k - j), {j, 0, k}]; Table[b[n, k], {n, 1, 10}, {k, 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<7, {7, 42, 210, 840, 2520, 5040, 5040}[[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, k):=n!/k!*x^(7*k-n)*sum(binomial(k, j)*binomial(7*j, n)*(-1)^(k-j), j, 0, k);

Crossrefs

Cf. A188062, A068424 (row 7).

Sequence in context: A176090 A183064 A290045 * A225327 A102532 A297405

Adjacent sequences: A188063 A188064 A188065 * A188067 A188068 A188069

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 24 2011

Status

approved