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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.
4
1, 4, 15, 55, 200, 726, 2639, 9620, 35190, 129200, 476102, 1760673, 6533150, 24319050, 90795375, 339929880, 1275977670, 4801199400, 18106714050, 68430306750, 259129680264, 983085703116, 3736124441990, 14222020085880, 54221213973500
OFFSET
1,2
LINKS
FORMULA
G.f.: (x + x^2*C^3)*C^3 where C = g.f. for Catalan numbers A000108.
E.g.f.: exp(2*x)*(Bessel_I(1,2*x)-Bessel_I(4,2*x)). - Paul Barry, Jun 04 2007
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
a(n) ~ 15 * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
From G. C. Greubel, Mar 18 2021: (Start)
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).
a(n) = binomial(2*n, n-1) - binomial(2*n, n-4) = A026009(2*n, n-1). (End)
MATHEMATICA
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 *)
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 *)
PROG
(Sage) [binomial(2*n, n-1) - binomial(2*n, n-4) for n in (1..30)] # G. C. Greubel, Mar 18 2021
(Magma) [Binomial(2*n, n-1) - Binomial(2*n, n-4): n in [1..30]]; // G. C. Greubel, Mar 18 2021
CROSSREFS
Sequence in context: A094833 A039717 A220948 * A371820 A050183 A094375
KEYWORD
nonn
EXTENSIONS
Corrected by Franklin T. Adams-Watters, Oct 25 2006
STATUS
approved