%I #47 Oct 28 2024 17:00:54
%S 1,7,91,1330,20475,324632,5245786,85900584,1420494075,23667689815,
%T 396704524216,6681687099710,112992892764570,1917283000904460,
%U 32626924340528840,556608279578340080,9516306085765295355,163011740982048945441
%N Binomial coefficient C(7n,n).
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%H T. D. Noe, <a href="/A004368/b004368.txt">Table of n, a(n) for n = 0..100</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F a(n) = C(7*n-1,n-1)*C(49*n^2,2)/(3*n*C(7*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014
%F G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002296. - _Vladimir Kruchinin_, Oct 05 2015
%F From _Ilya Gutkovskiy_, Jan 16 2017: (Start)
%F O.g.f.: 6F5(1/7,2/7,3/7,4/7,5/7,6/7; 1/6,1/3,1/2,2/3,5/6; 823543*x/46656).
%F E.g.f.: 6F6(1/7,2/7,3/7,4/7,5/7,6/7; 1/6,1/3,1/2,2/3,5/6,1; 823543*x/46656).
%F a(n) ~ sqrt(7/3)*7^(7*n)/(2*sqrt(Pi*n)*6^(6*n)). (End)
%F From _Peter Bala_, Feb 20 2022: (Start)
%F The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 6*A(x))^6 + (7^7)*x*A(x)^7 = 0.
%F Sum_{n >= 1} a(n)*( x*(6*x + 7)^6/(7^7*(1 + x)^7) )^n = x. (End)
%t Table[Binomial[7n,n],{n,0,20}] (* _Harvey P. Dale_, Apr 05 2014 *)
%o (Maxima)
%o B(x):=sum(binomial(7*n,n-1)/n*x^n,n,1,30);
%o taylor(x*diff(B(x),x)/B(x),x,0,10); /* _Vladimir Kruchinin_, Oct 05 2015 */
%o (PARI) a(n) = binomial(7*n,n) \\ _Altug Alkan_, Oct 05 2015
%o (Magma) [Binomial(7*n,n): n in [0..20]]; // _Vincenzo Librandi_, Oct 06 2015
%Y binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).
%Y Cf. A002296.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_