A106391 A "cosh transform" of binomial(2n,n-1).
0, 1, 4, 18, 80, 365, 1692, 7945, 37648, 179595, 861020, 4143832, 20004096, 96810779, 469508340, 2281123530, 11100465216, 54093131147, 263929559436, 1289217255934, 6303934406640, 30853639964847, 151139139048084
Offset: 0
Programs
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Maple
A106491 := proc(n) add(binomial(n, 2*k)*binomial(2*(n-2*k), n-2*k+1),k=0..floor(n/2)) ; end proc: # R. J. Mathar, Feb 20 2015
Formula
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)}.
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
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
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