A369088 - OEIS

%I #11 Jan 29 2024 06:05:22

%S 1,1,4,28,280,3639,57996,1093795,23821104,588282772,16243898516,

%T 495894495629,16584179388232,602955889304341,23678788166350620,

%U 998882687260157956,45047554811998482016,2162775743390757357579,110136661581764181626660,5929361362606879245799055,336484778280758295928357240

%N Expansion of Sum_{n>=0} ( (1+x)^n/(1-x)^n - (1-x)^n/(1+x)^n )^n / 4^n.

%H Paul D. Hanna, <a href="/A369088/b369088.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = Sum_{n>=0} ( (1+x)^n/(1-x)^n - (1-x)^n/(1+x)^n )^n / 4^n.

%F (2) A(x) = Sum_{n>=0} ((1+x)/(1-x))^(n^2)/4^n * Sum_{k=0..n} (-1)^k * binomial(n,k) * ((1-x)/(1+x))^(2*n*k).

%F a(n) ~ c * d^n * n^n, where d = 1.0710130838356321768944119175659886... and c = 0.80953649272682852412550716575... - _Vaclav Kotesovec_, Jan 29 2024

%e G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3639*x^5 + 57996*x^6 + 1093795*x^7 + 23821104*x^8 + 588282772*x^9 + 16243898516*x^10 + ...

%e By definition, A(x) equals the sum

%e A(x) = 1 + B_1(x) + B_2(x)^2 + B_3(x)^3 + B_4(x)^4 + ... + B_n(x)^n + ...

%e where

%e B_n(x) = ( (1+x)^n/(1-x)^n - (1-x)^n/(1+x)^n )/4,

%e explicitly,

%e B_1(x) = x + x^3 + x^5 + x^7 + x^9 + ...

%e B_2(x) = 2*x + 6*x^3 + 10*x^5 + 14*x^7 + 18*x^9 + ...

%e B_3(x) = 3*x + 19*x^3 + 51*x^5 + 99*x^7 + 163*x^9 + ...

%e B_4(x) = 4*x + 44*x^3 + 180*x^5 + 476*x^7 + 996*x^9 + ...

%e B_5(x) = 5*x + 85*x^3 + 501*x^5 + 1765*x^7 + 4645*x^9 + ...

%e B_6(x) = 6*x + 146*x^3 + 1182*x^5 + 5418*x^7 + 17718*x^9 + ...

%e B_7(x) = 7*x + 231*x^3 + 2471*x^5 + 14407*x^7 + 57799*x^9 + ...

%e ...

%e and

%e B_2(x)^2 = 4*x^2 + 24*x^4 + 76*x^6 + 176*x^8 + 340*x^10 + ...

%e B_3(x)^3 = 27*x^3 + 513*x^5 + 4626*x^7 + 26974*x^9 + 116901*x^11 + ...

%e B_4(x)^4 = 256*x^4 + 11264*x^6 + 231936*x^8 + 3005440*x^10 + ...

%e B_5(x)^5 = 3125*x^5 + 265625*x^7 + 10596875*x^9 + 265509375*x^11 + ...

%e B_6(x)^6 = 46656*x^6 + 6811776*x^8 + 469530432*x^10 + ...

%e B_7(x)^7 = 823543*x^7 + 190238433*x^9 + 20868579620*x^11 + ...

%e ...

%o (PARI) {a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0, n, ((1+X)^m/(1-X)^m - (1-X)^m/(1+X)^m)^m/4^m ); polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A319947, A319466, A369089.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 28 2024