login
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) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.
3

%I #17 Mar 23 2021 02:52:53

%S 1,8,45,219,987,4248,17748,72675,293436,1172908,4653935,18366075,

%T 72186075,282861360,1105877880,4316224860,16825024134,65525448960,

%U 255024693434,992116674142,3858537980286,15004402265424,58343871881400

%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) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.

%H G. C. Greubel, <a href="/A026015/b026015.txt">Table of n, a(n) for n = 3..1000</a>

%F -(n+6)*(n-3)*a(n) +2*(3*n^2+3*n-20)*a(n-1) +(-9*n^2+15*n+20)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) = 0. - _R. J. Mathar_, Jun 20 2013

%F From _G. C. Greubel_, Mar 19 2021: (Start)

%F G.f.: (1-x)*((1-3*x)*(1 -6*x +9*x^2 -3*x^3) -(1-x)*(1 -6*x +9*x^2 -x^3)*sqrt(1-4*x))/(2*x^6).

%F G.f.: (1-x)*x^3*C(x)^9, where C(x) is the g.f. of the Catalan numbers (A000108).

%F E.g.f.: exp(2*x)*(BesselI(3, 2*x) - BesselI(6, 2*x)).

%F a(n) = binomial(2*n, n-3) - binomial(2*n, n-6) = A026009(2*n, n-3).

%F a(n) = f(n) - f(n-1), where f(n) = Sum_{j=0..n-3} C(n-j-3)*(C(j+7) -6*C(j+6) +10*C(j+5) -4*C(j+4)) and C(n) are the Catalan numbers. (End)

%F From _G. C. Greubel_, Mar 22 2021: (Start)

%F a(n) = C(n+5) -8*C(n+4) +22*C(n+3) -25*C(n+2) +11*C(n+1) -C(n).

%F a(n) = (9/20)*(binomial(n,3)/binomial(n+6,5))*(3*n^2 +3*n +20)*C(n). (End)

%t Table[Binomial[2*n, n-3] - Binomial[2*n, n-6], {n, 3, 30}] (* _G. C. Greubel_, Mar 19 2021 *)

%o (Sage) [binomial(2*n, n-3) - binomial(2*n, n-6) for n in (3..30)] # _G. C. Greubel_, Mar 19 2021

%o (Magma) [Binomial(2*n, n-3) - Binomial(2*n, n-6): n in [3..30]]; // _G. C. Greubel_, Mar 19 2021

%Y First differences of A001392.

%Y Cf. A000108, A026009.

%K nonn

%O 3,2

%A _Clark Kimberling_