A190726 Central coefficients of Riordan matrix A118384.

1, 6, 62, 720, 8806, 110916, 1423796, 18520788, 243289670, 3220011684, 42872967012, 573608356272, 7705343534716, 103857425975400, 1403902871946000, 19024773303675420, 258372666772083270, 3515644245559211172, 47918193512409831380

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:

Lim_{n->oo} A316911(n)/A316912(n)/a(n) = log(2).

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

Cf. A118384, A123178.

Log(2) approximation rationals: A316911, A316912.

Cf. A123178.

Sequence in context: A342959 A144063 A186670 * A121251 A352071 A347016

Adjacent sequences: A190723 A190724 A190725 * A190727 A190728 A190729

Keyword

nonn

Author

Emanuele Munarini, May 17 2011

Status

approved