|
%I
%S 7,42,49,210,882,343,840,11172,12348,2401,2520,117600,288120,144060,
%T 16807,5040,1076040,5433120,5330220,1512630,117649,5040,8643600,
%U 89029080,155296680,81177810,14823774,823543,0,60540480,1306912320,3884433840,3360055440,1087076760,138355224,5764801
%N Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7,42,210,840,2520,5040,5040).
%C 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.
%H M. Abbas, S. Bouroubi, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.023">On new identities for Bell's polynomials</a>, Disc. Math 293. (1-3) (2005) 5-10
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065
%H John Riordan, <a href="http://dx.doi.org/10.1090/S0002-9904-1946-08621-8">Derivatives of composite functions</a>, Bull. Am. Math. Soc. 52 (1946) 664
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>, MathWorld.
%F B(n,k)=(n!/k!)*sum_{j=0..k} binomial(k,j) *binomial(7*j,n) *(-1)^(k-j).
%e 7;
%e 42,49;
%e 210,882,343;
%e 840,11172,12348,2401;
%e 2520,117600,288120,144060,16807;
%e 5040,1076040,5433120,5330220,1512630,117649;
%p A188066 := proc(n,k) n!/k!*add( binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j=0..k) ; end proc:
%p seq(seq(A188066(n,k),k=1..n),n=1..5) ; # R. J. Mathar, Apr 08 2011
%t 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 *)
%o (Maxima)
%o B(n,k):=n!/k!*x^(7*k-n)*sum(binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j,0,k);
%Y Cf. A188062, A068424 (row 7).
%K nonn,tabl
%O 1,1
%A _Vladimir Kruchinin_, Mar 24 2011
|
