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

Author

Vladimir Kruchinin, Mar 24 2011

Keywords

• nonn
• tabl

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.

Examples

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

Links

• 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.

Crossrefs

Cf. A188062, A068424 (row 7).

Programs

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 *)

Maxima

B(n,k):=n!/k!*x^(7*k-n)*sum(binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j,0,k);

Formula

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