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%I #39 Oct 01 2019 02:58:18
%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 and 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 [math.CO], 2011.
%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), 664-667.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F B(n,k) = (n!/k!)*Sum_{j=0..k} binomial(k,j)*binomial(7*j,n)*(-1)^(k-j).
%e Triangle begins
%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;
%e ...
%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
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> `if`(n<7,[7,42,210,840,2520,5040,5040][n+1],0), 9); # _Peter Luschny_, Jan 29 2016
%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 *)
%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 12;
%t B = BellMatrix[Function[n, If[n<7, {7, 42, 210, 840, 2520, 5040, 5040}[[n + 1]], 0]], rows];
%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%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
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