A163455 a(n) = binomial(5*n-1,n).

1, 4, 36, 364, 3876, 42504, 475020, 5379616, 61523748, 708930508, 8217822536, 95722852680, 1119487075980, 13136858812224, 154603005527328, 1824010149372864, 21566576904406820, 255485622301674660, 3031718514166879020, 36030431772522503316

Offset

0,2

Comments

Also, number of terms in A163142 with n zeros in binary representation.

All terms >= 4 are divisible by 4.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Formula

a(n) = (5n-1)!/(n!(4n-1)!).

G.f.: A(x)=x*B'(x)/B(x), where B(x)/x is g.f. for A118971. Also a(n) = Sum_{k=0..n} (binomial(n-1,n-k)*binomial(4*n,k)). - Vladimir Kruchinin, Oct 06 2015

From Peter Bala, Feb 14 2024: (Start)

a(n) = (-1)^n * binomial(-4*n, n).

a(n) = hypergeom([1 - 4*n, -n], [1], 1).

A(x) satisfies A(x/(1 + x)^5) = 1/(1 - 4*x). (End)

Example

a(1)=4 because there are 4 terms in A163142 with 1 zero in binary representation {23,27,29,30}_10 ={10111,11011,11101,11110}_2
a(2)=36 because there are 36 terms in A163142 with 2 zeros in binary representation: {639,703,735,751,759,763,765,766,831,863,879,887,891,893,894,927,943,951,955,957,958,975,983,987,989,990,999,1003,1005,1006,1011,1013,1014,1017,1018,1020}_10={1001111111,...,1111111100}_2
a(3)=364 terms in A163142 from 18431 to 32760 with 3 zeros in binary representation 18431_10=100011111111111_2 and 32760_10=111111111111000_2
a(4)=3876 terms in A163142 from 557055 to 1048560 with 4 zeros in binary representation, etc.

Mathematica

Table[(5*n-1)!/ n!/(4*n-1)!, {n, 20}]
Table[Binomial[5 n - 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

Prog

(Magma) [Binomial(5*n-1, n): n in [0..30]]; // Vincenzo Librandi, Aug 07 2014
(Maxima)
B(x):=sum(binomial(5*n-2, n-1)/(n)*x^n, n, 1, 30);
taylor(x*diff(B(x), x, 1)/B(x), x, 0, 10);
(PARI) a(n) = binomial(5*n-1, n); \\ Michel Marcus, Oct 06 2015

Crossrefs

Cf. A163142, A118971.

Sequence in context: A026334 A247562 A168595 * A371772 A138736 A266093

Adjacent sequences: A163452 A163453 A163454 * A163456 A163457 A163458

Keyword

nonn,base,easy

Author

Zak Seidov, Jul 28 2009

Extensions

Entry revised by N. J. A. Sloane, Dec 07 2015

Status

approved