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A112019 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k)^2.

%I A112019 #44 Apr 30 2025 09:11:27
%S A112019 1,5,55,749,11251,178835,2949115,49906925,860905315,15071939255,
%T A112019 266982872905,4774722189275,86070844191775,1561948324845095,
%U A112019 28507384046515555,522867506128197869,9631571375362268515,178094411589895650815,3304192479145474141741,61487420580006795749999
%N A112019 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k)^2.
%C A112019 Diagonal of rational function 1/(1 - x - y - z - x*y + x*z + x*y*z). - _Gheorghe Coserea_, Jul 01 2018
%C A112019 Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - z). - _Seiichi Manyama_, Apr 30 2025
%H A112019 Vincenzo Librandi, <a href="/A112019/b112019.txt">Table of n, a(n) for n = 0..200</a>
%H A112019 C. Elsner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., 43,1 (2005), 31-45.
%H A112019 V. 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 A112019 a(n) = 3F2( {-n, 1 + n, 1 + n} ; {1, 1} )(-1). - _Olivier Gérard_, Apr 23 2009
%F A112019 a(n) ~ (1+r)^(4*n+5/2)/r^(3*n+5/2)/(2*Pi*n)*sqrt((1-r)/(3-r)), where r is positive real root of the equation (1-r)*(1+r)^2=r^3, r = 1/6*((44-3*sqrt(177))^(1/3)+(44+3*sqrt(177))^(1/3)-1) = 0.82948354095849... - _Vaclav Kotesovec_, Nov 04 2012
%F A112019 Recurrence: 2*n^2*(59*n - 83)*a(n) = (2301*n^3 - 5538*n^2 + 3797*n - 800)*a(n-1) + 5*(59*n^3 - 201*n^2 + 213*n - 64)*a(n-2) + (59*n - 24)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, Nov 04 2012
%F A112019 G.f. y=A(x) satisfies: 0 = x*(5*x + 8)*(x^3 + 5*x^2 + 39*x - 2)*y'' + (15*x^4 + 82*x^3 + 315*x^2 + 624*x - 16)*y' + (5*x^3 + 21*x^2 + 80)*y. - _Gheorghe Coserea_, Jul 01 2018
%p A112019 seq(add((multinomial(n+k,n-k,k,k))*binomial(n+k,k),k=0..n),n=0..19); # _Zerinvary Lajos_, Oct 18 2006
%p A112019 ogf := hypergeom([1/12,5/12],[1], -1728*(x^3+5*x^2+39*x-2)*x^4 / (x^4+4*x^3+30*x^2-20*x+1)^3 ) / (x^4+4*x^3+30*x^2-20*x+1)^(1/4);
%p A112019 series(ogf, x=0, 30); # _Mark van Hoeij_, Jan 22 2013
%t A112019 Table[HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1, 1}, -1], {n, 0, 20}] (* _Olivier Gérard_, Apr 23 2009 *)
%t A112019 Table[Sum[Binomial[n,k]*Binomial[n+k,k]^2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 04 2012 *)
%o A112019 (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)^2); \\ _Michel Marcus_, Mar 09 2016
%Y A112019 Cf. A218693, A111968, A014178, A014180, A218689, A218692.
%Y A112019 Cf. A383536, A383537, A383538.
%K A112019 nonn
%O A112019 0,2
%A A112019 _N. J. A. Sloane_, Nov 28 2005