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%I #12 Oct 06 2020 05:07:44
%S 1,1,4,27,256,3152,47680,854802,17711872,416372620,10947581056,
%T 318304921165,10140097538560,351219420860694,13141237470041536,
%U 528208859187285899,22698715714385041920,1038485165851106374784,50395972495225521776384,2585595617532863164095240,139835798146777767415142912,7950987913261988583226011167
%N G.f.: Sum_{n>=0} ( (1+x)^n - (1-x)^n )^n / 2^n.
%H Paul D. Hanna, <a href="/A302108/b302108.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: Sum_{n>=0} [ Sum_{k=0..[n/2]} binomial(n,2*k+1) * x^(2*k+1) ]^n.
%F G.f.: Sum_{n>=0} (1+x)^(n^2) * Sum_{k=0..n} (-1)^k * C(n,k) * ((1-x)/(1+x))^(n*k) / 2^n.
%F a(n) ~ c * 2^(2*n) * n^n / (3^n * exp(n) * log(2)^(2*n)), where c = 0.873241746441310441203224293323899407211809744132... - _Vaclav Kotesovec_, Oct 06 2020
%e G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3152*x^5 + 47680*x^6 + 854802*x^7 + 17711872*x^8 + 416372620*x^9 + 10947581056*x^10 + ...
%e such that
%e A(x) = 1 + ((1+x) - (1-x))/2 + ((1+x)^2 - (1-x)^2)^2/2^2 + ((1+x)^3 - (1-x)^3)^3/2^3 + ((1+x)^4 - (1-x)^4)^4/2^4 + ((1+x)^5 - (1-x)^5)^5/2^5 + ((1+x)^6 - (1-x)^6)^6/2^6 + ((1+x)^7 - (1-x)^7)^7/2^7 + ...
%e Equivalently,
%e A(x) = 1 + x + (2*x)^2 + (3*x + x^3)^3 + (4*x + 4*x^3)^4 + (5*x + 10*x^3 + x^5)^5 + (6*x + 20*x^3 + 6*x^5)^6 + (7*x + 35*x^3 + 21*x^5 + x^7)^7 + (8*x + 56*x^3 + 56*x^5 + 8*x^7)^8 + (9*x + 84*x^3 + 126*x^5 + 36*x^7 + x^9)^9 + (10*x + 120*x^3 + 252*x^5 + 120*x^7 + 10*x^9)^10 + ...
%o (PARI) {a(n) = my(A=1); A = sum(m=0, n, ((1+x)^m - (1-x)^m +x*O(x^n))^m/2^m ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A301309, A301308.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 01 2018
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