OFFSET
0,2
REFERENCES
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.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
FORMULA
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
G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002296. - Vladimir Kruchinin, Oct 05 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
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).
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).
a(n) ~ sqrt(7/3)*7^(7*n)/(2*sqrt(Pi*n)*6^(6*n)). (End)
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 6*A(x))^6 + (7^7)*x*A(x)^7 = 0.
Sum_{n >= 1} a(n)*( x*(6*x + 7)^6/(7^7*(1 + x)^7) )^n = x. (End)
a(n) = A004369(n)*(6*n+1)/n. - R. J. Mathar, Mar 14 2026
D-finite with recurrence 72*n*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -7*(7*n-3)*(7*n-6)*(7*n-2)*(7*n-5)*(7*n-1)*(7*n-4)*a(n-1)=0. - R. J. Mathar, Mar 14 2026
MATHEMATICA
Table[Binomial[7n, n], {n, 0, 20}] (* Harvey P. Dale, Apr 05 2014 *)
PROG
(Maxima)
B(x):=sum(binomial(7*n, n-1)/n*x^n, n, 1, 30);
taylor(x*diff(B(x), x)/B(x), x, 0, 10); /* Vladimir Kruchinin, Oct 05 2015 */
(PARI) a(n) = binomial(7*n, n) \\ Altug Alkan, Oct 05 2015
(Magma) [Binomial(7*n, n): n in [0..20]]; // Vincenzo Librandi, Oct 06 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved