A004368 Binomial coefficient C(7n,n).

1, 7, 91, 1330, 20475, 324632, 5245786, 85900584, 1420494075, 23667689815, 396704524216, 6681687099710, 112992892764570, 1917283000904460, 32626924340528840, 556608279578340080, 9516306085765295355, 163011740982048945441

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 [alternative scanned copy].

Crossrefs

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).

Cf. A002296.

Programs

Magma

[Binomial(7*n,n): n in [0..20]]; // Vincenzo Librandi, Oct 06 2015

Mathematica

Table[Binomial[7n,n],{n,0,20}] (* Harvey P. Dale, Apr 05 2014 *)

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

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)