A063419 - OEIS

%I #57 Dec 09 2021 04:32:00

%S 1,6,146,4332,135954,4395456,144840476,4836766584,163112472594,

%T 5542414273884,189456975899496,6507792553644256,224442843729333276,

%U 7766945604528200460,269557528994032024080,9378595792117360310832

%N Central sextinomial coefficients.

%C Largest coefficient of (Sum_{j=0..5} x^j)^(2*n). a(n)= A018901(2*n).

%C The exponent of 2 in prime factorization of a(n) for n=1,2,...,7 is in A023520. - _Roman Witula_, Oct 07 2012

%C Central coefficients in triangle A063260. - _Zagros Lalo_, Sep 25 2018

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 605, 606.

%D R. Witula and D. Slota, Central trinomial coefficients and convolution type identities, Congressus Numerantium 201 (2010), 109-126.

%H G. C. Greubel, <a href="/A063419/b063419.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) =  A063260(2*n, 5*n)= [x^(5*n)](Sum_{j=0..5} x^j)^(2*n).

%F a(n) = Sum_{k=0..floor(5*n/6)} ((-1)^(k)*binomial(2*n,k)*binomial(7*n-6*k-1, 2*n-1)). - _Warut Roonguthai_, May 22 2006

%F 2*Pi*a(n) = Integral[-Pi;Pi] (sin(6*x)/sin(x))^(2*n) dx. The proof of this fact is in the Witula/Slota paper. - _Roman Witula_, Oct 07 2012

%F The Almkvist-Zeilberger algorithm in EKHAD establishes the following recurrence: -31104*(2*n+5)*(2*n+3)*(2*n+1)*(7*n+19)*(5*n+11)*(7*n+20)*(7*n+13)*(n+2)*(n+1)*a(n)+ 864*(7*n+20)*(2*n+5)*(2*n+3)*(n+2)*(25480*n^5+ 223496*n^4+755066*n^3+1223233*n^2+946889*n+279936)*a(n+1)- 6*(5*n+6)*(2*n+5)*(7*n+6)*(499359*n^6+ 6777015*n^5+38079431*n^4+113390385*n^3+18872398*n^2+ 166469280*n+60800544)*a(n+2)+ 5*(5*n+14)*(5*n+13)*(5*n+12)*(7*n+12)*(5*n+11)*(5*n+6)*(7*n+13)*(7*n+6)*(n+3)*a(n+3) = 0. - _Doron Zeilberger_, Apr 02 2013

%F a(n) ~ sqrt(3) * 36^n / sqrt(35*Pi*n). - _Vaclav Kotesovec_, Dec 09 2021

%t Table[Sum[(-1)^k*Binomial[2*n, k]*Binomial[7*n - 6*k - 1, 2*n - 1], {k, 0, Floor[5*n/6]}], {n, 0, 50}] (* _G. C. Greubel_, Mar 02 2017 *)

%o (PARI) concat([1], for(n=1,25, print1(sum(k=0,floor(5*n/6),(-1)^(k)*binomial(2*n,k)* binomial(7*n-6*k-1, 2*n-1)), ", "))) \\ _G. C. Greubel_, Mar 02 2017

%o (GAP) Concatenation([1],List([1..15],n->Sum([0..Int(5*n/6)],k->(-1)^k*Binomial(2*n,k)*Binomial(7*n-6*k-1,2*n-1)))); # _Muniru A Asiru_, Sep 26 2018

%Y Central q-nomial coefficients (appearing once) for q=2..5: A000984, A002426, A005721, A005191. For q=7: A025012.

%Y Cf. A063260.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 24 2001