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A188066 Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7, 42, 210, 840, 2520, 5040, 5040). 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 27 11:47 EDT 2022. Contains 354097 sequences. (Running on oeis4.)