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 A369089 #10 Jan 29 2024 06:13:22
%S A369089 2,52,17324,16145844,29572623276,89637821913140,406104654882450092,
%T A369089 2570553924485112780724,21667036447652685648382124,
%U A369089 234610018567852042809723552820,3173541919429028069213132538924460,52444911598298125255750037883777795508,1039685774587940760247676247641790182818220
%N A369089 Expansion of Sum_{n>=0} ( (1+x)^n/(1-x)^n + (1-x)^n/(1+x)^n )^n / 4^n, listing the coefficients of only the even powers of x.
%H A369089 Paul D. Hanna, <a href="/A369089/b369089.txt">Table of n, a(n) for n = 0..100</a>
%F A369089 G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) satisfies the following formulas.
%F A369089 (1) A(x) = Sum_{n>=0} ( (1+x)^n/(1-x)^n + (1-x)^n/(1+x)^n )^n / 4^n.
%F A369089 (2) A(x) = Sum_{n>=0} ((1+x)/(1-x))^(n^2)/4^n * Sum_{k=0..n} binomial(n,k) * ((1-x)/(1+x))^(2*n*k).
%F A369089 a(n) ~ c * d^n * n^(2*n), where d = 20.292093791236868074187146977890421458428... and c = 2.69404457993978919483089942207811854788... - _Vaclav Kotesovec_, Jan 29 2024
%e A369089 G.f. A(x) = 2 + 52*x^2 + 17324*x^4 + 16145844*x^6 + 29572623276*x^8 + 89637821913140*x^10 + 406104654882450092*x^12 + ...
%e A369089 By definition, A(x) equals the sum
%e A369089 A(x) = 1 + B_1(x)/2 + B_2(x)^2/2^2 + B_3(x)^3/2^3 + ... + B_n(x)^n/2^n + ...
%e A369089 where B_n(x) is the even bisection of (1+x)^n/(1-x)^n,
%e A369089 B_n(x) = ( (1+x)^n/(1-x)^n + (1-x)^n/(1+x)^n )/2;
%e A369089 explicitly,
%e A369089 B_1(x) = 1 + 2*x^2 + 2*x^4 + 2*x^6 + 2*x^8 + 2*x^10 + ...
%e A369089 B_2(x) = 1 + 8*x^2 + 16*x^4 + 24*x^6 + 32*x^8 + 40*x^10 + ...
%e A369089 B_3(x) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + 258*x^8 + 402*x^10 + ...
%e A369089 B_4(x) = 1 + 32*x^2 + 192*x^4 + 608*x^6 + 1408*x^8 + 2720*x^10 + ...
%e A369089 B_5(x) = 1 + 50*x^2 + 450*x^4 + 1970*x^6 + 5890*x^8 + 14002*x^10 + ...
%e A369089 B_6(x) = 1 + 72*x^2 + 912*x^4 + 5336*x^6 + 20256*x^8 + 58728*x^10 + ...
%e A369089 B_7(x) = 1 + 98*x^2 + 1666*x^4 + 12642*x^6 + 59906*x^8 + 209762*x^10 + ...
%e A369089 ...
%e A369089 and
%e A369089 B_2(x)^2 = 1 + 16*x^2 + 96*x^4 + 304*x^6 + 704*x^8 + ...
%e A369089 B_3(x)^3 = 1 + 54*x^2 + 1170*x^4 + 13398*x^6 + 93762*x^8 + ...
%e A369089 B_4(x)^4 = 1 + 128*x^2 + 6912*x^4 + 207232*x^6 + 3868160*x^8 + ...
%e A369089 B_5(x)^5 = 1 + 250*x^2 + 27250*x^4 + 1709850*x^6 + 69024450*x^8 + ...
%e A369089 B_6(x)^6 = 1 + 432*x^2 + 83232*x^4 + 9466896*x^6 + 710899776*x^8 + ...
%e A369089 B_7(x)^7 = 1 + 686*x^2 + 213346*x^4 + 39887470*x^6 + 5019056770*x^8 + ...
%e A369089 ...
%o A369089 (PARI) /* Print only the coefficients of the even powers of x in A(x) */
%o A369089 {a(n) = my(A=1, X=x + x*O(x^(2*n))); A=suminf(m=0,((1+X)^m/(1-X)^m + (1-X)^m/(1+X)^m)^m/4^m*1. ); round(polcoeff(A,2*n))}
%o A369089 for(n=0,20, print1(a(n),", "))
%Y A369089 Cf. A369088.
%K A369089 nonn
%O A369089 0,1
%A A369089 _Paul D. Hanna_, Jan 28 2024