cp's OEIS Frontend

A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

%I A359643 #25 Apr 17 2025 09:34:55
%S A359643 1,5,37,317,2885,27105,259765,2523813,24768069,244941833,2437083697,
%T A359643 24367722725,244639635749,2464477467769,24899468129405,
%U A359643 252202062544617,2560119328830725,26038134699958233,265278657849511561,2706809063101138409,27657194997231516145,282941098708193905485
%N A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).
%C A359643 In general, for m>1, Sum_{k=0..n} binomial(n,k) * binomial(m*k,k) ~ sqrt((m + (1 - 1/m)^(m-1))/(m-1)) * (1 + m^m/(m-1)^(m-1))^n / sqrt(2*Pi*n).
%H A359643 Robert Israel, <a href="/A359643/b359643.txt">Table of n, a(n) for n = 0..977</a>
%F A359643 a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)).
%F A359643 Conjecture D-finite with recurrence +81*n*(3*n-1)*(3*n-2)*a(n) +3*(243*n^3-8433*n^2+14984*n-7064)*a(n-1) +2*(-58607*n^3+297306*n^2-491401*n+269124)*a(n-2) +6*(n-2)*(56663*n^2-237722*n+252221)*a(n-3) -3*(n-2)*(n-3)*(111625*n-286402)*a(n-4) +110653*(n-2)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jan 09 2023
%F A359643 a(n) = 4F3( -n,1/4,1/2,3/4 ; 1/3, 2/3,1 ; -256/27). - _R. J. Mathar_, Jan 10 2023
%F A359643 a(n) = [x^n] (1 + 5*x + 6*x^2 + 4*x^3 + x^4)^n. - _Ilya Gutkovskiy_, Apr 17 2025
%p A359643 A359643 := proc(n)
%p A359643     hypergeom([-n,1/4,1/2,3/4],[1/3,2/3,1],-256/27) ;
%p A359643     simplify(%) ;
%p A359643 end proc:
%p A359643 seq(A359643(n),n=0..40) ; # _R. J. Mathar_, Jan 10 2023
%t A359643 Table[Sum[Binomial[n, k]*Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
%o A359643 (PARI) a(n) = sum(k=0, n, binomial(n,k) * binomial(4*k,k)); \\ _Michel Marcus_, Jan 09 2023
%Y A359643 Cf. A026375, A188686.
%Y A359643 Cf. A156887, A346646, A346664.
%K A359643 nonn
%O A359643 0,2
%A A359643 _Vaclav Kotesovec_, Jan 09 2023