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A112029 a(n) = Sum_{k=0..n} binomial(n+k, k)^2.

%I A112029 #47 Mar 21 2025 09:40:20
%S A112029 1,5,46,517,6376,82994,1119210,15475205,217994860,3115374880,
%T A112029 45035696036,657153097330,9663914317396,143050882063262,
%U A112029 2129448324373546,31853280798384645,478503774600509620,7215090439396842572,109154411037070011504,1656268648035559711392
%N A112029 a(n) = Sum_{k=0..n} binomial(n+k, k)^2.
%H A112029 Vincenzo Librandi, <a href="/A112029/b112029.txt">Table of n, a(n) for n = 0..200</a>
%H A112029 F. Baldassarri, S. Bosch, B. Dwork, (eds), <a href="https://doi.org/10.1007/BFb0091139">p-adic Analysis. Lecture Notes in Mathematics</a>, vol. 1454, pp. 194 - 204,  Springer, Berlin, Heidelberg.
%H A112029 Matthijs J. Coster, <a href="https://core.ac.uk/download/pdf/301642554.pdf">Supercongruences</a>.
%H A112029 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 A112029 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
%H A112029 Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, <a href="http://arxiv.org/abs/1602.04347">Sums of powers of Catalan triangle numbers</a>, arXiv:1602.04347 [math.NT], 2016.
%F A112029 a(n) ~ 2^(4*n+2)/(3*Pi*n). - _Vaclav Kotesovec_, Nov 23 2012
%F A112029 Recurrence: 2*(2*n+1)*(21*n-13)*n^2*a(n) = (1365*n^4 - 1517*n^3 + 240*n^2 + 216*n - 64)*a(n-1) - 4*(n-1)*(2*n-1)^2*(21*n+8)*a(n-2). - _Vaclav Kotesovec_, Nov 23 2012
%F A112029 G.f.: see Maple code. - _Mark van Hoeij_, Mar 27 2013
%F A112029 a(p-1) == 1 (mod p^3) for all primes p >= 5. See the comments in A173774. - _Peter Bala_, Jul 12 2024
%F A112029 a(n-1) = 1/(4*n) * binomial(2*n, n)^2 * ( 1 + 3*((n - 1)/(n + 1))^3 + 5*((n - 1)*(n - 2)/((n + 1)*(n + 2)))^3 + 7*((n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)))^3 + ... )  for n >= 1. - _Peter Bala_, Jul 22 2024
%F A112029 a(m*p^r - 1) == a(m*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and positive integers m and r. See Coster, Theorem 4. - _Peter Bala_, Nov 29 2024
%F A112029 a(n) = A110197(2n,n). - _Alois P. Heinz_, Mar 21 2025
%p A112029 f := 64*x^2/(16*x-1); S := sqrt(x)*sqrt(4-x);
%p A112029 H := ((10*x-5/8)*hypergeom([1/4,1/4],[1],f)-(21*x-21/8)*hypergeom([1/4,5/4],[1],f))/(S*(1-16*x)^(5/4));
%p A112029 ord := 30;
%p A112029 ogf := series(int(series(H,x=0,ord),x)/S,x=0,ord);
%p A112029 # _Mark van Hoeij_, Mar 27 2013
%t A112029 Table[Sum[Binomial[n+k,k]^2, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Nov 23 2012 *)
%o A112029 (Magma) [(&+[Binomial(n+j, j)^2: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Jul 06 2021
%o A112029 (Sage) [sum(binomial(n+j, j)^2 for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jul 06 2021
%o A112029 (PARI) a(n) = sum(k=0, n, binomial(n+k, k)^2); \\ _Michel Marcus_, Jul 07 2021
%Y A112029 Cf. A001700, A110197, A112028, A173774, A176335, A219562, A219563, A219564, A333592.
%K A112029 nonn,easy
%O A112029 0,2
%A A112029 _N. J. A. Sloane_, Nov 28 2005