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 A248167 #29 Jun 02 2025 09:52:08
%S A248167 1,49,3249,261121,23512801,2266426449,228110356881,23642146057761,
%T A248167 2502698427758529,269194720423487089,29319711378381802609,
%U A248167 3225762406810715071041,357859427246543331576481,39977637030683399494792849,4492572407488016429783217489,507445676088537643607528136801
%N A248167 Expansion of g.f.: 1 / AGM(1-33*x, sqrt((1-9*x)*(1-121*x))).
%C A248167 In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by 1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*( Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k) )^2, and consists of integer coefficients when 4|(q-p). Here AGM(x, y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
%H A248167 Seiichi Manyama, <a href="/A248167/b248167.txt">Table of n, a(n) for n = 0..481</a>
%F A248167 a(n) = A248168(n)^2 = ( Sum_{k=0..n} 3^(n-k)*2^k * C(n,k) * C(2*k,k) )^2.
%F A248167 G.f.: 1 / AGM((1-3*x)*(1+11*x), (1+3*x)*(1-11*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F A248167 a(n) ~ 11^(2*n + 1) / (8*Pi*n). - _Vaclav Kotesovec_, Sep 27 2019
%e A248167 G.f.: A(x) = 1 + 49*x + 3249*x^2 + 261121*x^3 + 23512801*x^4 +...
%e A248167 where the square-root of the terms yields A248168:
%e A248167 [1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, ...],
%e A248167 the g.f. of which is 1/sqrt((1-3*x)*(1-11*x)).
%t A248167 a[n_] := Sum[3^(n - k) * 2^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n} ]^2; Array[a, 17, 0] (* _Amiram Eldar_, Dec 11 2018 *)
%o A248167 (PARI) {a(n,p=3,q=11)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
%o A248167 for(n=0, 20, print1(a(n,3,11), ", "))
%o A248167 (PARI) {a(n,p=3,q=11)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
%o A248167 for(n=0, 20, print1(a(n,3,11), ", "))
%o A248167 (PARI) {a(n,p=3,q=11)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
%o A248167 for(n=0, 20, print1(a(n,3,11), ", "))
%o A248167 (Magma)
%o A248167 m:=40;
%o A248167 A248168:= [n le 2 select 7^(n-1) else (7*(2*n-3)*Self(n-1) - 33*(n-2)*Self(n-2))/(n-1) : n in [1..m+2]];
%o A248167 A248167:= func< n | (A248168[n+1])^2 >;
%o A248167 [A248167(n): n in [0..m]]; // _G. C. Greubel_, May 31 2025
%o A248167 (SageMath)
%o A248167 @CachedFunction
%o A248167 def b(n): # b = A248168
%o A248167 if (n<2): return 7^n
%o A248167 else: return (7*(2*n-1)*b(n-1) - 33*(n-1)*b(n-2))//n
%o A248167 def A248167(n): return (b(n))^2
%o A248167 print([A248167(n) for n in range(41)]) # _G. C. Greubel_, May 31 2025
%Y A248167 Cf. A084771, A246467, A246906, A246923, A248168.
%K A248167 nonn
%O A248167 0,2
%A A248167 _Paul D. Hanna_, Oct 03 2014