login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003518 a(n) = 8*binomial(2*n+1,n-3)/(n+5).
(Formerly M4529)
26
1, 8, 44, 208, 910, 3808, 15504, 62016, 245157, 961400, 3749460, 14567280, 56448210, 218349120, 843621600, 3257112960, 12570420330, 48507033744, 187187399448, 722477682080, 2789279908316, 10772391370048, 41620603020640 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

a(n-6) is the number of n-th generation nodes in the tree of sequences with unit increase labeled by 7 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003

Number of standard tableaux of shape (n+4,n-3). - Emeric Deutsch, May 30 2004

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 3..500

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

H. H. Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A. Vol. 21 (2010), No. 2, pp. 265-284 (see 4.5 p. 280).

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

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.

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

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

O. Mandelshtam, Multi-Catalan tableaux and the two-species TASEP, arXiv:1502.00948 [math.CO], 2015.

O. Mandelshtam, Multi-Catalan tableaux and the two-species TASEP, Ann. Inst. Henri Poincaré Comb. Phys. Interact. vol. 3 (2016), 321-348, DOI 10.4171/AIHPD/30.

L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90.

L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]

_Zoran Sunic_, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.

FORMULA

G.f.: x^3*C(x)^8, where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004

The convolution of A002057 with itself. - Gerald McGarvey, Nov 08 2007

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>=7, a(n-4)=(-1)^(n-7)*coeff(charpoly(A,x),x^7). - Milan Janjic, Jul 08 2010

a(n) = A214292(2*n,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012

Integral representation as the n-th moment of the signed weight function W(x) on (0,4), i.e. in Maple notation: a(n+3) = int(x^n*W(x),x=0..4), n=0,1..., with W(x) = (1/2)*x^(7/2)*(x-2)*(x^2-4*x+2)*sqrt(4-x)/Pi. - Karol A. Penson , Oct 26 2016

From Ilya Gutkovskiy, Jan 22 2017: (Start)

E.g.f.: 4*BesselI(4,2*x)*exp(2*x)/x.

a(n) ~ 4^(n+2)/(sqrt(Pi)*n^(3/2)). (End)

EXAMPLE

G.f. = x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...

MATHEMATICA

Table[8 Binomial[2 n + 1, n - 3]/(n + 5), {n, 3, 25}] (* Michael De Vlieger, Oct 26 2016 *)

CoefficientList[Series[((1 - Sqrt[1 - 4 x])/(2 x))^8, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 23 2017 *)

PROG

(PARI) {a(n) = if( n<3, 0, 8 * binomial(2*n + 1, n-3) / (n + 5))}; /* Michael Somos, Mar 14 2011 */

(PARI) x='x+O('x^50); Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^8) \\ Altug Alkan, Nov 01 2015

(MAGMA) [8*Binomial(2*n+1, n-3)/(n+5): n in [3..30]]; // Vincenzo Librandi, Jan 23 2017

CROSSREFS

Cf. A002057.

First differences are in A026018.

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003519.

Sequence in context: A270627 A273639 A022636 * A100575 A272112 A271005

Adjacent sequences:  A003515 A003516 A003517 * A003519 A003520 A003521

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jon E. Schoenfield, May 06 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 13 19:20 EST 2017. Contains 295976 sequences.