|
%I #22 Mar 23 2021 02:52:45
%S 1,6,28,119,483,1911,7448,28764,110466,422807,1615152,6163885,
%T 23514855,89714835,342411120,1307613480,4997082510,19111589280,
%U 73154916744,280265589198,1074685552094,4124573481446,15843809385168,60914041121640
%N a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 6. Also a(n) = T(2n,n-2), where T is the array defined in A026009.
%H G. C. Greubel, <a href="/A026014/b026014.txt">Table of n, a(n) for n = 2..1000</a>
%F -(n-2)*(n+5)*(n+23)*a(n) +(-n^3+127*n^2+188*n-432)*a(n-1) +2*(n-1)*(2*n-3)*(5*n-24)*a(n-2) = 0. - _R. J. Mathar_, Jun 20 2013
%F From _G. C. Greubel_, Mar 19 2021: (Start)
%F G.f.: (1-x)*(1 -7*x +14*x^2 -7*x^3 -(1 -5*x +6*x^2 -x^3)*sqrt(1-4*x))/(2*x^5).
%F G.f.: (1-x)*x^2*C(x)^7, where C(x) is the g.f. of the Catalan numbers (A000108).
%F E.g.f.: exp(2*x)*(BesselI(2, 2*x) - BesselI(5, 2*x)).
%F a(n) = binomial(2*n, n-2) - binomial(2*n, n-5) = A026009(2*n, n-2).
%F a(n) = 1 if n = 2 else f(n) - f(n-1), where f(n) = Sum_{j=0..n-2} C(n-j-2)*(C(j+5) -4*C(j+4) +3*C(j+3)) and C(n) are the Catalan numbers. (End)
%F From _G. C. Greubel_, Mar 22 2021: (Start)
%F a(n) = C(n+4) -6*C(n+3) +11*C(n+2) -7*C(n+1) +C(n).
%F a(n) = 21*((n*(n-1)*(n^2+n+4)/((n+2)*(n+3)*(n+4)*(n+5)))*C(n), where C(n) are the Catalan numbers. (End)
%t Table[Binomial[2*n, n-2] - Binomial[2*n, n-5], {n, 2, 30}] (* _G. C. Greubel_, Mar 19 2021 *)
%o (Sage) [binomial(2*n, n-2) - binomial(2*n, n-5) for n in (2..30)] # _G. C. Greubel_, Mar 19 2021
%o (Magma) [Binomial(2*n, n-2) - Binomial(2*n, n-5): n in [2..30]]; // _G. C. Greubel_, Mar 19 2021
%Y First differences of A000588.
%Y Cf. A000108, A026009.
%K nonn
%O 2,2
%A _Clark Kimberling_
|
