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A106391 A "cosh transform" of binomial(2n,n-1).

%I A106391 #5 Feb 20 2015 17:28:01
%S A106391 0,1,4,18,80,365,1692,7945,37648,179595,861020,4143832,20004096,
%T A106391 96810779,469508340,2281123530,11100465216,54093131147,263929559436,
%U A106391 1289217255934,6303934406640,30853639964847,151139139048084
%N A106391 A "cosh transform" of binomial(2n,n-1).
%C A106391 Mean of binomial and inverse binomial transform of A001791.
%F A106391 E.g.f.: cosh(x)exp(2x)I_1(2x), where I_1 is Bessel function; a(n)=sum{k=0..floor(n/2), binomial(n, 2k)binomial(2(n-2k), n-2k+1)}.
%F A106391 Conjecture: -(n+1)*(n-2)*a(n) +4*n*(3*n-7)*a(n-1) +(-49*n^2+193*n-148)*a(n-2) +8*(9*n-19)*(n-4)*a(n-3) +(5*n^2+115*n-418)*a(n-4) -12*(7*n-19)*(n-4)*a(n-5) +45*(n-4)*(n-5)*a(n-6)=0. - _R. J. Mathar_, Feb 20 2015
%F A106391 Conjecture: -(n-2)*(n+1)*(4*n^2-16*n+19)*a(n) +8*n*(n-2)*(4*n^2-14*n+13)*a(n-1) +2*(-28*n^4+168*n^3-375*n^2+401*n-180)*a(n-2) -8*(n-2)*(4*n^3-18*n^2+27*n-10)*a(n-3) +15*(n-2)*(n-3)*(4*n^2-8*n+7)*a(n-4)=0. - _R. J. Mathar_, Feb 20 2015
%p A106391 A106491 := proc(n)
%p A106391     add(binomial(n, 2*k)*binomial(2*(n-2*k), n-2*k+1),k=0..floor(n/2)) ;
%p A106391 end proc: # _R. J. Mathar_, Feb 20 2015
%K A106391 easy,nonn
%O A106391 0,3
%A A106391 _Paul Barry_, May 01 2005