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%I #57 Sep 26 2022 05:42:33
%S 1,12,90,544,2907,14364,67298,303600,1332045,5722860,24192090,
%T 100975680,417225900,1709984304,6962078952,28192122176,113649492522,
%U 456442180920,1827459250276,7297426411968,29075683360185,115631433392020,459124809056550,1820529677650320,7210477496434485
%N a(n) = 12 * C(2n+1,n-5) / (n+7).
%C Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.
%C Number of standard tableaux of shape (n+6,n-5). - _Emeric Deutsch_, May 30 2004
%H G. C. Greubel, <a href="/A090749/b090749.txt">Table of n, a(n) for n = 5..1000</a>
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/1707.09918">Bounce statistics for rational lattice paths</a>, arXiv:1707.09918 [math.CO], 2017, p. 9.
%H Richard K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>.
%H Richard K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.
%F a(n) = A039598(n, 5) = A033184(n+7, 12).
%F G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
%F Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - _Milan Janjic_, Jul 08 2010
%F a(n) = A214292(2*n,n-6) for n > 5. - _Reinhard Zumkeller_, Jul 12 2012
%F From _Karol A. Penson_, Nov 21 2016: (Start)
%F O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
%F Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
%F Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
%F Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
%F From _Ilya Gutkovskiy_, Nov 21 2016: (Start)
%F E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
%F a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
%F From _Amiram Eldar_, Sep 26 2022: (Start)
%F Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)
%t Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* _G. C. Greubel_, Feb 07 2017 *)
%o (PARI) for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ _G. C. Greubel_, Feb 07 2017
%Y Cf. A001622, A009766, A039598, A059365, A214292.
%Y Cf. A000108, A002057, A003517, A003518, A003519.
%K easy,nonn
%O 5,2
%A _Philippe Deléham_, Feb 15 2004
%E Missing term 113649492522 inserted by _Ilya Gutkovskiy_, Dec 07 2016
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