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%I #28 Jul 18 2018 17:37:20
%S 1,6,62,720,8806,110916,1423796,18520788,243289670,3220011684,
%T 42872967012,573608356272,7705343534716,103857425975400,
%U 1403902871946000,19024773303675420,258372666772083270,3515644245559211172,47918193512409831380
%N Central coefficients of Riordan matrix A118384.
%C This sequence gives the integer part of an integral approximation to log(2), thus bears strong similarity to A123178. Quality of rational approximants appears entirely sufficient to prove irrationality. - _Bradley Klee_, Jun 29 2018
%H Vincenzo Librandi, <a href="/A190726/b190726.txt">Table of n, a(n) for n = 0..88</a>
%H Wadim Zudilin, <a href="https://arxiv.org/abs/math/0404523"> An essay on irrationality measures of pi and other logarithms</a>, arXiv:math/0404523 [math.NT], 2004.
%F a(n) = T(2*n,n), where T(n,k) = A118384(n,k).
%F a(n) = Sum_{k=0..n} binomial(2*n, k)*binomial(2*n, n-k)*2^k.
%F a(n) = Sum_{k=0..n} binomial(2*n, k)*binomial(k, n-k)*2^(n-k)*3^(2*k-n).
%F From _Bradley Klee_, Jun 29 2018: (Start)
%F a(n)*log(2) - A316911(n)/A316912(n) = I_n = Integral_{t=0..1}(-1/2)^n/(1+t)*((1-t)^2*t^2/(1+t))^n*dt.
%F Lim_{n->oo} I_n = 0, therefore:
%F Lim_{n->oo} A316911(n)/A316912(n)/a(n) = log(2).
%F G.f. G(x) and derivatives G^(n)(x) = d^n/dx^n G(x) satisfy a Picard-Fuchs type differential equation, 0 = Sum_{m=0..5,n=0..3} M_{m,n} x^m G^(n)(x), with integer matrix: M = {{324,-54,0,0}, {-36,10842,-486,0}, {84,8352,14931,-243}, {0,756,19026,3024}, {0,0,672,5364}, {0,0,0,112}}.
%F 2*(n-1)*(2*n-3)*(2*n-1)*(33*n-8)*a(n-2)+ 9*(2*n-1)*(693*n^3-1554*n^2+989*n-160)*a(n-1) -3*n*(3*n-2)*(3*n-1)*(33*n-41)*a(n)=0.
%F (End)
%e From _Bradley Klee_, Jul 16 2018: (Start)
%e I_2 = Integral_{t=0..1} ((1-t)^4*t^4)/(4*(1+t)^3)*dt = 62*log(2) - 1719/40 < 10^(-3).
%e I_3 = Integral_{t=0..1} - ((1-t)^6*t^6)/(8*(1+t)^4)*dt = 720*log(2) - 143731/288 < 10^(-5). (End)
%t Table[Sum[Binomial[2n,k]Binomial[2n,n-k]2^k,{k,0,n}],{n,0,100}]
%t RecurrenceTable[{2*(n-1)*(2*n-3)*(2*n-1)*(33*n-8)*a[n-2]+ 9*(2*n-1)*(693*n^3-1554*n^2+989*n-160)*a[n-1] -3*n*(3*n-2)*(3*n-1)*(33*n-41)*a[n]==0, a[0]==1,a[1]==6},a,{n,0,10}] (* _Bradley Klee_, Jun 29 2018 *)
%o (Maxima) makelist(sum(binomial(2*n,k)*binomial(2*n,n-k)*2^k,k,0,n),n,0,12);
%o (PARI) a(n)=sum(k=0,n,binomial(2*n,k)*binomial(2*n,n-k)<<k) \\ _Charles R Greathouse IV_, Jun 29 2011
%Y Cf. A118384, A123178.
%Y Log(2) approximation rationals: A316911, A316912.
%Y Cf. A123178.
%K nonn
%O 0,2
%A _Emanuele Munarini_, May 17 2011
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