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A072287 Let f(n, m) = binomial(n - m/2 + 1, n - m + 1) - binomial(n - m/2, n - m + 1) and let s(n) = Sum_{k=0..n} f(n, k); then a(n) = numerator of s(n).

%I A072287 #8 Feb 06 2019 17:20:03
%S A072287 1,2,7,47,155,2027,6597,42835,138875,3599155,11654465,75457289,
%T A072287 244238477,3161900479,10232916665,66231885067,214336798299,
%U A072287 11097918730051,35913975952793,232441522435405,752199270651129
%N A072287 Let f(n, m) = binomial(n - m/2 + 1, n - m + 1) - binomial(n - m/2, n - m + 1) and let s(n) = Sum_{k=0..n} f(n, k); then a(n) = numerator of s(n).
%H A072287 Robert Israel, <a href="/A072287/b072287.txt">Table of n, a(n) for n = 0..1234</a>
%F A072287 s(0)=1, s(1)=2, s(n+1)=s(n)+s(n-1)+binomial(n-1/2, n) for n>=1.
%F A072287 (2*n+3)*s(n) - s(n+1) + (-4*n-7)*s(n+2) + (2*n+4)*s(n+3) = 0. - _Robert Israel_, Feb 06 2019
%e A072287 1,2,7/2,47/8,155/16,2027/128,6597/256,42835/1024,138875/2048,...
%p A072287 S:= gfun:-rectoproc({s(0)=1,s(1)=2,s(2)=7/2,(2*n+3)*s(n) - s(n+1) + (-4*n-7)*s(n+2) + (2*n+4)*s(n+3) = 0, s(n),remember):
%p A072287 map(numer@S, [$0..30]); # _Robert Israel_, Feb 06 2019
%t A072287 f[n_, m_] := Binomial[n - m/2 + 1, n - m + 1] - Binomial[n - m/2, n - m + 1]; s[n_] := Sum[ f[n, k], {k, 0, n}]; Table [Numerator[s[n]], {n, 0, 26}]
%Y A072287 Denominator of s(n+1) = A046161(n).
%K A072287 nonn,easy,frac
%O A072287 0,2
%A A072287 Michele Dondi (bik.mido(AT)tiscalinet.it), Jul 11 2002