This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A168597 #51 May 13 2025 14:15:02
%S A168597 1,1,9,49,361,2601,19881,154449,1225449,9853321,80156209,658076409,
%T A168597 5444816521,45343869481,379735715529,3195538786449,27004932177129,
%U A168597 229066136374761,1949470542590481,16640188083903609,142415188146838161,1221800234100831441,10504959504381567729
%N A168597 Squares of the central trinomial coefficients (A002426).
%C A168597 Ignoring initial term, a(n) equals the logarithmic derivative of A168598.
%C A168597 Partial sums of A007987. Hence, a(n) is the number of irreducible words of length at most 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero. - _Max Alekseyev_, Jun 05 2011
%C A168597 The number of ways a king, starting at the origin of an infinite chessboard, can return to the origin in n moves, where leaving the king where it is counts as a move. Cf. A094061. - _Peter Bala_, Feb 14 2017
%H A168597 G. C. Greubel, <a href="/A168597/b168597.txt">Table of n, a(n) for n = 0..1000</a>
%F A168597 a(n) = A002426(n)^2.
%F A168597 G.f.: hypergeom([1/12, 5/12],[1],1728*x^4*(x-1)*(9*x-1)*(3*x+1)^2/(81*x^4-36*x^3-26*x^2-4*x+1)^3)/(81*x^4-36*x^3-26*x^2-4*x+1)^(1/4). - _Mark van Hoeij_, May 07 2013
%F A168597 G.f.: 1 / AGM(1+3*x, sqrt((1-x)*(1-9*x))), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - _Paul D. Hanna_, Sep 04 2014
%F A168597 G.f.: 1 / AGM((1-x)*(1-3*x), (1+x)*(1+3*x)) = Sum_{n>=0} a(n)*x^(2*n). - _Paul D. Hanna_, Oct 04 2014
%F A168597 a(n) = (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([(1-n)/2,-n/2],[1],4). - _Peter Luschny_, Nov 10 2014
%F A168597 a(n) ~ 3^(2*n+1) / (4*Pi*n). - _Vaclav Kotesovec_, Sep 28 2019
%F A168597 From _Peter Bala_, Feb 08 2022: (Start)
%F A168597 a(n) = Sum_{k = 0..n} (-3)^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
%F A168597 n^2*(2*n-3)*a(n)= (7*n^2-14*n+6)*(2*n-1)*a(n-1) + 3*(7*n^2-14*n+6)*(2*n-3)*a(n-2) - 27*(2*n-1)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = 9.
%F A168597 G.f.: A(x) = Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 3*x)^(2*n+1).
%F A168597 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k.
%F A168597 Conjecture: The stronger congruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and positive integers n and k. (End)
%F A168597 G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+3*x)^2)/(1+3*x). - _Mark van Hoeij_, May 13 2025
%p A168597 a := n -> (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([1/2-n/2,-n/2],[1], 4): seq(simplify(a(n)),n=0..20); # _Peter Luschny_, Nov 10 2014
%t A168597 Table[(-1)^n*Hypergeometric2F1[1/2, -n, 1, 4] * Hypergeometric2F1[(1 - n)/2, -n/2, 1, 4], {n, 0, 50}] (* _G. C. Greubel_, Feb 26 2017 *)
%t A168597 CoefficientList[Series[(2 EllipticK[(16 x)/(1 + 3 x)^2])/(Pi (1 + 3 x)), {x, 0, 28}], x, 26] (* After _Mark van Hoeij_, _Peter Luschny_, May 13 2025 *)
%o A168597 (PARI) {a(n)=polcoeff((1+x+x^2 +x*O(x^n))^n,n)^2}
%o A168597 for(n=0, 20, print1(a(n), ", "))
%o A168597 (PARI) /* Using AGM: */
%o A168597 {a(n)=polcoeff( 1 / agm(1+3*x, sqrt((1+3*x)^2 - 16*x +x*O(x^n))), n)}
%o A168597 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Sep 04 2014
%Y A168597 Cf. A002426, A133053, A168598, A243949, A094061.
%K A168597 nonn,easy
%O A168597 0,3
%A A168597 _Paul D. Hanna_, Dec 01 2009