A277178 a(n) = Sum_{k=0..n} k*binomial(2*k,k)/2.

0, 1, 7, 37, 177, 807, 3579, 15591, 67071, 285861, 1209641, 5089517, 21314453, 88918353, 369734553, 1533115953, 6341759073, 26177411943, 107853629643, 443633635743, 1822098923943, 7473806605563, 30618895206483, 125303348573883, 512274592771083, 2092407173242983, 8539348101568335

Offset: 0

Author

Vladimir Reshetnikov, Oct 02 2016

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1658
• Eric Weisstein's World of Mathematics, Central Binomial Coefficient.

Crossrefs

Row 3 of A323222.

Cf. A000984, A002457.

Programs

Maple

a:=n->sqrt(-1/27)-((n+1)/2)*binomial(2*(n+1),n+1)*hypergeom([1,n+3/2],[n+1],4):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Oct 03 2016

Mathematica

Table[Binomial[2 n, n] (2 n + 1 - Hypergeometric2F1[1, -n, 1/2 - n, 1/4])/3, {n, 0, 30}]

PARI

{a(n) = sum(k=0, n, k*binomial(2*k, k))/2} \\ Seiichi Manyama, Jan 29 2019

Formula

a(n) = binomial(2*n,n) * (2*n + 1 - hypergeom([1,-n], [1/2-n], 1/4))/3.

a(n+1) - a(n) = A002457(n) = (2*n+1)!/n!^2.

Recurrence: (5*n + 2) * a(n) = (4*n + 2) * a(n-1) + n * a(n+1).

a(n) ~ sqrt(n) * 2^(2*n+1) / (3*sqrt(Pi)). - Vaclav Kotesovec, Jan 29 2019

G.f.: x/(1-x) * (1-4*x)^(-3/2). - Seiichi Manyama, Jan 29 2019