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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).

%I A188066 #40 Feb 16 2025 08:33:14
%S A188066 7,42,49,210,882,343,840,11172,12348,2401,2520,117600,288120,144060,
%T A188066 16807,5040,1076040,5433120,5330220,1512630,117649,5040,8643600,
%U A188066 89029080,155296680,81177810,14823774,823543,0,60540480,1306912320,3884433840,3360055440,1087076760,138355224,5764801
%N A188066 Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7, 42, 210, 840, 2520, 5040, 5040).
%C A188066 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 A188066 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 A188066 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 A188066 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 A188066 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A188066 B(n,k) = (n!/k!)*Sum_{j=0..k} binomial(k,j)*binomial(7*j,n)*(-1)^(k-j).
%e A188066 Triangle begins
%e A188066      7;
%e A188066     42,      49;
%e A188066    210,     882,     343;
%e A188066    840,   11172,   12348,    2401;
%e A188066   2520,  117600,  288120,  144060,   16807;
%e A188066   5040, 1076040, 5433120, 5330220, 1512630, 117649;
%e A188066   ...
%p A188066 A188066 := proc(n,k) n!/k!*add( binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j=0..k) ; end proc:
%p A188066 seq(seq(A188066(n,k),k=1..n),n=1..5) ; # _R. J. Mathar_, Apr 08 2011
%p A188066 # The function BellMatrix is defined in A264428.
%p A188066 # Adds (1,0,0,0, ..) as column 0.
%p A188066 BellMatrix(n -> `if`(n<7,[7,42,210,840,2520,5040,5040][n+1],0), 9); # _Peter Luschny_, Jan 29 2016
%t A188066 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 A188066 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A188066 rows = 12;
%t A188066 B = BellMatrix[Function[n, If[n<7, {7, 42, 210, 840, 2520, 5040, 5040}[[n + 1]], 0]], rows];
%t A188066 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A188066 (Maxima)
%o A188066 B(n,k):=n!/k!*x^(7*k-n)*sum(binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j,0,k);
%Y A188066 Cf. A188062, A068424 (row 7).
%K A188066 nonn,tabl
%O A188066 1,1
%A A188066 _Vladimir Kruchinin_, Mar 24 2011