cp's OEIS Frontend

A026013 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) = 4. Also a(n) = T(2n,n-1), where T is the array defined in A026009.

%I A026013 #16 Mar 18 2021 22:59:53
%S A026013 1,4,15,55,200,726,2639,9620,35190,129200,476102,1760673,6533150,
%T A026013 24319050,90795375,339929880,1275977670,4801199400,18106714050,
%U A026013 68430306750,259129680264,983085703116,3736124441990,14222020085880,54221213973500
%N A026013 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) = 4. Also a(n) = T(2n,n-1), where T is the array defined in A026009.
%H A026013 G. C. Greubel, <a href="/A026013/b026013.txt">Table of n, a(n) for n = 1..1000</a>
%F A026013 G.f.: (x + x^2*C^3)*C^3 where C = g.f. for Catalan numbers A000108.
%F A026013 E.g.f.: exp(2*x)*(Bessel_I(1,2*x)-Bessel_I(4,2*x)). - _Paul Barry_, Jun 04 2007
%F A026013 Conjecture: (n+4)*(n+1)*a(n) -4*(n^2+4*n+6)*a(n-1) +4*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Nov 13 2012
%F A026013 a(n) ~ 15 * 4^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 03 2019
%F A026013 From _G. C. Greubel_, Mar 18 2021: (Start)
%F A026013 a(n) = C(n+3) - 4*C(n+2) + 3*C(n+1) + Sum_{j=0..n-1} C(j+1)*C(n-j-1), where C(n) are the Catalan numbers (A000108).
%F A026013 a(n) = binomial(2*n, n-1) - binomial(2*n, n-4) = A026009(2*n, n-1). (End)
%t A026013 Rest[CoefficientList[Series[(-1 + Sqrt[1 - 4*x])^3 * (1 - x) * (-1 + Sqrt[1 - 4*x] + 2*x) / (16*x^4), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Sep 03 2019 *)
%t A026013 With[{f = CatalanNumber}, Table[f[n+3] -4*f[n+2] +3*f[n+1] +Sum[f[j+1]*f[n-j-1], {j, 0, n-1}], {n,30}]] (* _G. C. Greubel_, Mar 18 2021 *)
%o A026013 (Sage) [binomial(2*n, n-1) - binomial(2*n, n-4) for n in (1..30)] # _G. C. Greubel_, Mar 18 2021
%o A026013 (Magma) [Binomial(2*n, n-1) - Binomial(2*n, n-4): n in [1..30]]; // _G. C. Greubel_, Mar 18 2021
%Y A026013 Cf. A000108, A026009.
%K A026013 nonn
%O A026013 1,2
%A A026013 _Clark Kimberling_
%E A026013 Corrected by _Franklin T. Adams-Watters_, Oct 25 2006