A101500 A Chebyshev transform of the central binomial numbers.

1, 2, 5, 16, 53, 178, 609, 2112, 7393, 26066, 92437, 329360, 1178149, 4228322, 15218305, 54907136, 198527617, 719170850, 2609577701, 9483269008, 34508808789, 125727351186, 458573578977, 1674270763584, 6118472289889, 22378379004146, 81913223571701

Offset

0,2

Comments

A Chebyshev transform of A000984. Under the Chebyshev transform, we map a g.f. g(x) to (1/(1+x^2))g(x/(1+x^2).

Also equal to the Riordan array (1/(1-x)^2,x/(1-x)^2) applied to aerated central binomial coefficients (with g.f. 1/sqrt(1-4x^2)). - Paul Barry, Jul 06 2009

Directed 2-D walks with n steps starting at (0,0) and ending on the X-axis using steps N,S,E,W and avoiding N followed by S. - David Scambler, Jun 24 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

C. Banderier and P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554 [The formula in Table 2 for A101500 is not correct], also arXiv:1201.6116

P. Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.

Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Formula

G.f.: 1/(sqrt(1+x^2)*sqrt(1-4*x+x^2)).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*C(2(n-2k), n-2k).

From Paul Barry, Jul 06 2009: (Start)

G.f.: 1/((1-x)^2-2*x^2/((1-x)^2-x^2/((1-x)^2-x^2/((1-x)^2-... (continued fraction);

a(n) = Sum_{k=0..n} C(n+k+1,n-k)*C(k,k/2)*(1+(-1)^k)/2. (End)

Conjecture: n*a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*a(n-2) +2*(-2*n+3)*a(n-3) +(n-2)*a(n-4)=0. - R. J. Mathar, Nov 16 2012

a(n) ~ sqrt(1/2 + 7/(8*sqrt(3))) * (2+sqrt(3))^n / sqrt(Pi*n). - Vaclav Kotesovec, Feb 08 2014

a(n) = Sum_{k=ceiling(n/2)..n} C(k,n-k)^2*2^(2*k-n). - Vladimir Kruchinin, Apr 09 2014

Mathematica

CoefficientList[Series[1/Sqrt[(1+x^2)*(1-4*x+x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
Table[1/2^n* Sum[(-1)^k*Binomial[2 k, k]* Sum[Binomial[n - 2 k, j]^2*3^j, {j, 0, n - 2 k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 30 2018 *)
Table[Sum[Binomial[n - k, k]*(-1)^k*Binomial[2 (n - 2 k), n - 2 k], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 30 2018 *)
a[ n_] := Sum[Binomial[n + k + 1, 2k + 1] Binomial[k, Quotient[k, 2]], {k, 0, n, 2}]; (* Michael Somos, Jun 30 2018 *)

Prog

(PARI) A101500(maxx)={n=0; while(n<=maxx, z=sum(k=0, floor(n/2), binomial(n-k, k)*binomial(2*(n-2*k), n-2*k)*(-1)^k ) ; print1(z, ", "); n+=1); } \\ Bill McEachen, Jan 02 2016
(PARI) x='x+O('x^40); Vec(1/(sqrt(1+x^2)*sqrt(1-4*x+x^2))); \\ Michel Marcus, Jan 03 2016

Crossrefs

Cf. A000984.

Sequence in context: A268571 A001428 A055726 * A339828 A047063 A268430

Adjacent sequences: A101497 A101498 A101499 * A101501 A101502 A101503

Keyword

nonn,easy

Author

Paul Barry, Dec 04 2004

Status

approved