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A218693
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a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)^3.
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5
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1, 9, 271, 11193, 535251, 27854739, 1531656211, 87547358649, 5149886133907, 309721191497259, 18957806551405701, 1177134132932168739, 73964787438524189871, 4694347514292389411103, 300499277330710307643771, 19378727805024033594228153, 1257802636907811605342461587
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OFFSET
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0,2
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LINKS
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FORMULA
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Recurrence: 4*(n-1)*n^3*(29412*n^4 - 224352*n^3 + 632931*n^2 - 781692*n + 356309)*a(n) = 2*(n-1)*(4176504*n^7 - 38122740*n^6 + 140783586*n^5 - 270139161*n^4 + 288226505*n^3 - 170040251*n^2 + 51625509*n - 6283008)*a(n-1) + 2*(1647072*n^8 - 19152000*n^7 + 94636812*n^6 - 258386460*n^5 + 423728203*n^4 - 423743982*n^3 + 249392728*n^2 - 77793627*n + 9736704)*a(n-2) + 2*(n-2)*(235296*n^7 - 2618352*n^6 + 11905158*n^5 - 28432149*n^4 + 38188669*n^3 - 28610816*n^2 + 10954716*n - 1618272)*a(n-3) - (n-2)*(29412*n^4 - 106704*n^3 + 136347*n^2 - 71238*n + 12608)*(n-3)^3*a(n-4).
a(n) ~ (1+r)^(6*n+7/2)/r^(4*n+7/2)/(4*Pi^(3/2)*n^(3/2))*sqrt((1-r)/(2-r)), where r is positive real root of the equation (1-r)*(1+r)^3=r^4, r = 0.90340819201887...
Generally, Sum_{k=0..n} C(n,k)^p*C(n+k,k)^q is asymptotic to sqrt((r*(1-r^2))/((p+q+(p-q)*r)*(2*Pi*n)^(p+q-1))) * (1+r)^(q*n)/(1-r)^(p*n+p), where r is positive real root of the equation (1-r)^p*(1+r)^q=r^(p+q). - Vaclav Kotesovec, Nov 07 2012
a(n) = hypergeom([-n, n+1, n+1, n+1],[1, 1, 1], -1). - Detlef Meya, May 25 2024
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MATHEMATICA
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Table[Sum[Binomial[n, k]*Binomial[n+k, k]^3, {k, 0, n}], {n, 0, 20}]
a[n_] := HypergeometricPFQ[{-n, n+1, n+1 , n+1}, {1, 1, 1}, -1]; Table[a[n], {n, 0, 16}] (* Detlef Meya, May 25 2024 *)
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CROSSREFS
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Cf.
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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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