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A002458 a(n) = binomial(4n+1, 2n). 9

%I

%S 1,10,126,1716,24310,352716,5200300,77558760,1166803110,17672631900,

%T 269128937220,4116715363800,63205303218876,973469712824056,

%U 15033633249770520,232714176627630544,3609714217008132870,56093138908331422716,873065282167813104916

%N a(n) = binomial(4n+1, 2n).

%D The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1982, (3.109), page 35.

%H T. D. Noe, <a href="/A002458/b002458.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = Sum_{k=0..n} 4^k * binomial( n + k, n) * binomial( 2*n - 2*k, n - k). - _Michael Somos_, Feb 25 2012

%F a(n) = A001700(2*n) = (n+1)*A000108(2*n+1).

%F G.f.: (4-(1+4*y)*c(y)-(1-4*y)*c(-y))/(2*(1-(4*y)^2)) with y^2=x, c(y)= g.f. for A000108 (Catalan). - _Wolfdieter Lang_, Dec 13 2001

%F a(n) ~ 2^(1/2)*Pi^(-1/2)*n^(-1/2)*2^(4*n)*{1 - 5/16*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002

%F a(n) = A024492(n)*(n+1). - _R. J. Mathar_, Aug 10 2015

%F G.f.: 2F1(3/4,5/4;3/2;16x). - _R. J. Mathar_, Aug 10 2015

%F n*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1) = 0. - _R. J. Mathar_, Aug 10 2015

%F From _Peter Bala_, Nov 04 2015: (Start)

%F a(n) = 4^n*binomial(2*n + 1/2,n).

%F O.g.f.: sqrt( c(4*x)/(1 - 16*x) ). In general, c(x)^k/sqrt(1 - 4*x) is the o.g.f. for the sequence binomial(2*n + k,n). (End)

%F From _Ilya Gutkovskiy_, Jan 17 2017: (Start)

%F E.g.f.: 2F2(3/4,5/4; 1,3/2; 16*x).

%F Sum_{n>=0} 1/a(n) = 3F2(1,1,3/2; 3/4,5/4; 1/16) = 1.108563435104316693... (End)

%F From _Peter Bala_, Mar 16 2018: (Start)

%F The right-hand side of the binomial coefficient identity Sum_{k = 0..n} 4^(n-k)*C(2*n+1,2*k)*C(2*k,k) = a(n).

%F a(n) = 4^n*hypergeom([-n,-n-1/2], [1], 1). (End)

%e 1 + 10*x + 126*x^2 + 1716*x^3 + 24310*x^4 + 352716*x^5 + 5200300*x^6 + ...

%p A002458:=n->binomial(4*n+1,2*n): seq(A002458(n), n=0..30); # _Wesley Ivan Hurt_, Jan 17 2017

%t Table[Binomial[4n+1,2n],{n,0,30}] (* _Harvey P. Dale_, Apr 04 2011 *)

%t 4^Range[0, 22] Simplify[ CoefficientList[ Series[ Sqrt[2]/(((Sqrt[1 - 4 x] + 1)^(1/2))*Sqrt[1 - 4 x]), {x, 0, 22}], x]] (* _Robert G. Wilson v_, Aug 08 2011 *)

%o (PARI) a(n) = binomial( 4*n + 1, 2*n)

%Y Cf. A000984, A001448, A001700, A024492.

%Y Row sums of A067001.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

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Last modified December 14 02:56 EST 2018. Contains 318087 sequences. (Running on oeis4.)