A317662 - OEIS

%I #13 Aug 09 2018 04:41:07

%S 1,2,24,448,11820,401392,16668960,818355488,46367354632,2977828665832,

%T 213763450387456,16961461169786752,1474091484740240064,

%U 139256465915227044352,14208358055857371300864,1557104405499802200814464,182416569911688799401148816,22749429746475540390909166048,3009102958766163591152586574464,420767787785916464100556297780608

%N G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 2^n  / (3 - 2*(1+x)^n)^(n+1).

%C The following identities hold for |y| <= 1 and fixed real k > 0:

%C (C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).

%C (C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1).

%C This sequence is an example of (C2) when y = 1+x and k = 2.

%H Vaclav Kotesovec, <a href="/A317662/b317662.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 2^n / (3 - 2*(1+x)^n)^(n+1).

%F (2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 2^n / (3 + 2*(1+x)^n)^(n+1).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = 7.5592435681748721825440151469382350654183499600538671407998439255608144356... and c = 0.30852178850187571906358489049387403704035769403106379389644818349... - _Vaclav Kotesovec_, Aug 09 2018

%e G.f.: A(x) = 1 + 2*x + 24*x^2 + 448*x^3 + 11820*x^4 + 401392*x^5 + 16668960*x^6 + 818355488*x^7 + 46367354632*x^8 + ...

%e such that

%e A(x) = 1  +  ((1+x) - 1)*2/(3 - 2*(1+x))^2  +  ((1+x)^2 - 1)^2*2^2/(3 - 2*(1+x)^2)^3  +  ((1+x)^3 - 1)^3*2^3/(3 - 2*(1+x)^3)^4  +  ((1+x)^4 - 1)^4*2^4/(3 - 2*(1+x)^4)^5  +  ((1+x)^5 - 1)^5*2^5/(3 - 2*(1+x)^5)^6  +  ((1+x)^6 - 1)^6*2^6/(3 - 2*(1+x)^6)^7 + ...

%e Also,

%e A(x) = 1/5  +  ((1+x) + 1)*2/(3 + 2*(1+x))^2  +  ((1+x)^2 + 1)^2*2^2/(3 + 2*(1+x)^2)^3  +  ((1+x)^3 + 1)^3*2^3/(3 + 2*(1+x)^3)^4  +  ((1+x)^4 + 1)^4*2^4/(3 + 2*(1+x)^4)^5  +  ((1+x)^5 + 1)^5*2^5/(3 + 2*(1+x)^5)^6  +  ((1+x)^6 + 1)^6*2^6/(3 + 2*(1+x)^6)^7 + ...

%e EXAMPLE OF SUMS.

%e Evaluating the g.f. formally at x = -1/2, we obtain the sums

%e S1 = Sum_{n>=0} (1 - 2^n)^n * 4^n / (3*2^n - 2)^(n+1),

%e S2 = Sum_{n>=0} (1 + 2^n)^n * 4^n / (3*2^n + 2)^(n+1),

%e explicitly,

%e S1 = 1 - 4/4^2 + 3^2*4^2/10^3 - 7^3*4^3/22^4 + 15^4*4^4/46^5 - 31^5*4^5/94^6 + 63^6*4^6/190^7 - 127^7*4^7/382^8 + 255^8*4^8/766^9 - 511^9*4^9/1534^10 + 1023^10*4^10/3070^11 + ...

%e S2 = 1/5 + 3*4/8^2 + 5^2*4^2/14^3 + 9^3*4^3/26^4 + 17^4*4^4/50^5 + 33^5*4^5/98^6 + 65^6*4^6/194^7 + 129^7*4^7/386^8 + 257^8*4^8/770^9 + 513^9*4^9/1538^10 + 1025^10*4^10/3074^11 + ...

%e where S1 = S2 = 0.8378452129227094466992700455568437913726753230322...

%o (PARI) {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 2^m / (3 - 2*(1+x)^m +x*O(x^n) )^(m+1) ); ;polcoeff(A,n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A302598, A317663, A317664, A302614, A317350.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 03 2018