A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset

5,2

Comments

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

Links

G. C. Greubel, Table of n, a(n) for n = 5..1000

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.

Richard K. Guy, Letter to N. J. A. Sloane, May 1990.

Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Formula

a(n) = A039598(n, 5) = A033184(n+7, 12).

G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).

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

a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012

From Karol A. Penson, Nov 21 2016: (Start)

O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.

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.

Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).

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).

From Ilya Gutkovskiy, Nov 21 2016: (Start)

E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.

a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)

From Amiram Eldar, Sep 26 2022: (Start)

Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).

Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Mathematica

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n, 5, 50}] (* G. C. Greubel, Feb 07 2017 *)

Prog

(PARI) for(n=5, 50, print1(12*binomial(2*n+1, n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

Crossrefs

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Sequence in context: A073382 A036216 A022640 * A130592 A002544 A093801

Adjacent sequences: A090746 A090747 A090748 * A090750 A090751 A090752

Keyword

easy,nonn

Author

Philippe Deléham, Feb 15 2004

Extensions

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

Status

approved