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A218693 a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)^3.

%I A218693 #20 May 26 2024 13:40:57
%S A218693 1,9,271,11193,535251,27854739,1531656211,87547358649,5149886133907,
%T A218693 309721191497259,18957806551405701,1177134132932168739,
%U A218693 73964787438524189871,4694347514292389411103,300499277330710307643771,19378727805024033594228153,1257802636907811605342461587
%N A218693 a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)^3.
%H A218693 Vincenzo Librandi, <a href="/A218693/b218693.txt">Table of n, a(n) for n = 0..200</a>
%H A218693 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012
%F A218693 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).
%F A218693 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...
%F A218693 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
%F A218693 a(n) = hypergeom([-n, n+1, n+1, n+1],[1, 1, 1], -1). - _Detlef Meya_, May 25 2024
%t A218693 Table[Sum[Binomial[n,k]*Binomial[n+k,k]^3,{k,0,n}],{n,0,20}]
%t A218693 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 *)
%Y A218693 Cf.
%Y A218693 A001850 (p=1, q=1),
%Y A218693 A112019 (p=1, q=2),
%Y A218693 A005258 (p=2, q=1),
%Y A218693 A005259 (p=2, q=2),
%Y A218693 A111968 (p=2, q=3),
%Y A218693 A014178 (p=3, q=1),
%Y A218693 A014180 (p=3, q=2),
%Y A218693 A092813 (p=3, q=3),
%Y A218693 A218690 (p=4, q=2),
%Y A218693 A092814 (p=4, q=4),
%Y A218693 A092815 (p=5, q=5),
%Y A218693 A218692 (p=6, q=3),
%Y A218693 A218689 (p=6, q=6).
%K A218693 nonn
%O A218693 0,2
%A A218693 _Vaclav Kotesovec_, Nov 04 2012