OFFSET
0,2
COMMENTS
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..88
Wadim Zudilin, An essay on irrationality measures of pi and other logarithms, arXiv:math/0404523 [math.NT], 2004.
FORMULA
a(n) = T(2*n,n), where T(n,k) = A118384(n,k).
a(n) = Sum_{k=0..n} binomial(2*n, k)*binomial(2*n, n-k)*2^k.
a(n) = Sum_{k=0..n} binomial(2*n, k)*binomial(k, n-k)*2^(n-k)*3^(2*k-n).
From Bradley Klee, Jun 29 2018: (Start)
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.
Lim_{n->oo} I_n = 0, therefore:
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}}.
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.
(End)
EXAMPLE
From Bradley Klee, Jul 16 2018: (Start)
I_2 = Integral_{t=0..1} ((1-t)^4*t^4)/(4*(1+t)^3)*dt = 62*log(2) - 1719/40 < 10^(-3).
I_3 = Integral_{t=0..1} - ((1-t)^6*t^6)/(8*(1+t)^4)*dt = 720*log(2) - 143731/288 < 10^(-5). (End)
MATHEMATICA
Table[Sum[Binomial[2n, k]Binomial[2n, n-k]2^k, {k, 0, n}], {n, 0, 100}]
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 *)
PROG
(Maxima) makelist(sum(binomial(2*n, k)*binomial(2*n, n-k)*2^k, k, 0, n), n, 0, 12);
(PARI) a(n)=sum(k=0, n, binomial(2*n, k)*binomial(2*n, n-k)<<k) \\ Charles R Greathouse IV, Jun 29 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, May 17 2011
STATUS
approved