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A277178
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a(n) = Sum_{k=0..n} k*binomial(2*k,k)/2.
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2
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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).
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MAPLE
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a:=n->sqrt(-1/27)-((n+1)/2)*binomial(2*(n+1), n+1)*hypergeom([1, n+3/2], [n+1], 4):
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MATHEMATICA
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Table[Binomial[2 n, n] (2 n + 1 - Hypergeometric2F1[1, -n, 1/2 - n, 1/4])/3, {n, 0, 30}]
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PROG
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(PARI) {a(n) = sum(k=0, n, k*binomial(2*k, k))/2} \\ Seiichi Manyama, Jan 29 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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