A101500 - OEIS

%I #59 Jan 17 2025 06:41:02

%S 1,2,5,16,53,178,609,2112,7393,26066,92437,329360,1178149,4228322,

%T 15218305,54907136,198527617,719170850,2609577701,9483269008,

%U 34508808789,125727351186,458573578977,1674270763584,6118472289889,22378379004146,81913223571701

%N A Chebyshev transform of the central binomial numbers.

%C 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)).

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A101500/b101500.txt">Table of n, a(n) for n = 0..1000</a>

%H C. Banderier and P. Hitczenko, <a href="http://doi.org/10.1016/j.dam.2011.12.011">Enumeration and asymptotics of restricted compositions having the same number of parts</a>, Disc. Appl. Math. 160 (18) (2012) 2542-2554 [The formula in Table 2 for A101500 is not correct], also <a href="https://arxiv.org/abs/1201.6116">arXiv:1201.6116</a>

%H P. Barry, <a href="http://dx.doi.org/10.1155/2013/657806">On the Connection Coefficients of the Chebyshev-Boubaker polynomials</a>, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.

%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

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

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

%F From Paul Barry, Jul 06 2009: (Start)

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

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

%F 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

%F a(n) ~ sqrt(1/2 + 7/(8*sqrt(3))) * (2+sqrt(3))^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 08 2014

%F a(n) = Sum_{k=ceiling(n/2)..n} C(k,n-k)^2*2^(2*k-n). - _Vladimir Kruchinin_, Apr 09 2014

%t CoefficientList[Series[1/Sqrt[(1+x^2)*(1-4*x+x^2)], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)

%t 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 *)

%t 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 *)

%t a[ n_] := Sum[Binomial[n + k + 1, 2k + 1] Binomial[k, Quotient[k, 2]], {k, 0, n, 2}]; (* _Michael Somos_, Jun 30 2018 *)

%o (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

%o (PARI) x='x+O('x^40); Vec(1/(sqrt(1+x^2)*sqrt(1-4*x+x^2))); \\ _Michel Marcus_, Jan 03 2016

%Y Cf. A000984.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Dec 04 2004