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A001791 a(n) = binomial coefficient C(2n, n-1).
(Formerly M3500 N1421)
115
0, 1, 4, 15, 56, 210, 792, 3003, 11440, 43758, 167960, 646646, 2496144, 9657700, 37442160, 145422675, 565722720, 2203961430, 8597496600, 33578000610, 131282408400, 513791607420, 2012616400080, 7890371113950, 30957699535776, 121548660036300, 477551179875952 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of peaks at even level in all Dyck paths of semilength n+1. Example: a(2)=4 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even level are shown by *. - Emeric Deutsch, Dec 05 2003

Also number of long ascents (i.e., ascents of length at least two) in all Dyck paths of semilength n+1. Example: a(2)=4 because in the five Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and (UUU)DDD, we have four long ascents (shown between parentheses). Here U=(1,1) and D=(1,-1). Also number of branch nodes (i.e., vertices of outdegree at least two) in all ordered trees with n+1 edges. - Emeric Deutsch, Feb 22 2004

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=1. Example: For n=2 these are the paths EENN, ENEN, ENNE and NEEN. - Herbert Kociemba, May 23 2004

Narayana transform (A001263) of [1, 3, 5, 7, 9, ...] = (1, 4, 15, 56, 210, ...). Row sums of triangles A136534 and A136536. - Gary W. Adamson, Jan 04 2008

Starting with offset 1 = the Catalan sequence starting (1, 2, 5, 14, ...) convolved with A000984: (1, 2, 6, 20, ...). - Gary W. Adamson, May 17 2009

Also number of peaks plus number of valleys in all Dyck n-paths. - David Scambler, Oct 08 2012

Apparently counts UDDUD in all Dyck paths of semilength n+2. - David Scambler, Apr 22 2013

Apparently the number of peaks strictly left of the midpoint in all Dyck paths of semilength n+1. - David Scambler, Apr 30 2013

For n>0, a(n) is the number of compositions of n into at most n parts if zeros are allowed as parts (so-called "weak" compositions). - L. Edson Jeffery, Jul 24 2014

Number of paths in the half-plane x >= 0, from (0,0) to (2n,2), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 4 paths: UUUD, UUDU, UDUU, DUUU. - José Luis Ramírez Ramírez, Apr 19 2015

For n>1, 1/a(n) is the probability that when a stick is broken up at n points independently and uniformly chosen at random along its length any triple of pieces of the n+1 pieces can form a triangle. The corresponding probability for the existence of at least one triple is A339392(n)/A339393(n). - Amiram Eldar, Dec 04 2020

a(n) is the number of lattice paths of 2n steps taken from the step set {U=(1,1), D=(1,-1)} that start at the origin, never go below the x-axis, and end strictly above the x-axis; more succinctly, proper left factors of Dyck paths. For example, a(2)=4 counts UUUU, UUUD, UUDU, UDUU. - David Callan and Emeric Deutsch, Jan 25 2021

From Gus Wiseman, Jul 21 2021: (Start)

Also the number of integer compositions of 2n+1 with alternating sum -1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(1) = 1 through a(3) = 15 compositions are:

  (1,2)  (2,3)      (3,4)

         (1,3,1)    (1,4,2)

         (1,1,1,2)  (2,4,1)

         (1,2,1,1)  (1,1,2,3)

                    (1,2,2,2)

                    (1,3,2,1)

                    (2,1,1,3)

                    (2,2,1,2)

                    (2,3,1,1)

                    (1,1,1,3,1)

                    (1,2,1,2,1)

                    (1,3,1,1,1)

                    (1,1,1,1,1,2)

                    (1,1,1,2,1,1)

                    (1,2,1,1,1,1)

The following relate to these compositions.

- The unordered version is A000070.

- Allowing any negative alternating sum gives A000346.

- The opposite (positive 1) version is A000984.

- The version for reverse-alternating sum is also A001791 (this sequence).

- Taking alternating sum -2 instead of -1 gives A002054.

- The shifted version for alternating sum 0 is counted by A088218 and ranked by A344619.

- Ranked by A345910 (reverse: A345912).

Equivalently, a(n) counts binary numbers with 2n+1 digits and one more 0 than 1's. For example, the a(2) = 4 binary numbers are: 10001, 10010, 10100, 11000.

(End)

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Cornelius Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

T. D. Noe and Matuszka Tamás, Table of n, a(n) for n = 0..1200 (terms n = 0..200 from T. D. Noe)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2550-2558; preprint, 2016.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.

Miklós Bóna, Surprising Symmetries in Objects Counted by Catalan Numbers, Electronic J. Combin., 19 (2012), P62, eq. (6).

Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.

Guo-Niu Han, Enumeration of Standard Puzzles.

Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]

A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 260-288.

Milan Janjic, Two Enumerative Functions.

Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.

Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.

Christian Krattenthaler and Daniel Yaqubi, Some determinants of path generating functions, II, Advances in Applied Mathematics, Vol. 101 (2018), pp. 232-265; arXiv preprint, arXiv:1802.05990 [math.CO], 2018.

Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)

Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3. - From N. J. A. Sloane, Sep 16 2012

Zhi Lan Wang, Tautological integrals on symmetric products of curves, Acta Mathematica Sinica, English Series, Vol. 32, No. 8 (2016), pp. 901-910; arXiv preprint, arXiv:1506.08405 [math.AG], 2015-2016; alternative link.

Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

FORMULA

a(n) = n*A000108(n).

G.f.: x*(d/dx)c(x) where c(x) = Catalan g.f. - Wolfdieter Lang

Convolution of A001700 (central binomial of odd order) and A000108 (Catalan): a(n+1) = Sum_{k=0..n} C(k)*binomial(2*(n-k)+1, n-k), C(k): Catalan. - Wolfdieter Lang

E.g.f.: exp(2x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 08 2002

a(n) = Sum_{k=0..n} C(n, k)*C(n, k+1). - Paul Barry, May 15 2003

a(n) = Sum_{i=1..n} binomial(i+n-1, n).

G.f.: (1-2x-sqrt(1-4x))/(2x*sqrt(1-4x)). - Emeric Deutsch, Dec 05 2003

a(n) = A092956/(n!). - Amarnath Murthy, Jun 16 2004

a(n) = binomial(2n,n) - A000108(n). - Paul Barry, Apr 21 2005

a(n) = (1/(2*Pi))*Integral_{x=0..4} (x^n*(x-2)/sqrt(x(4-x))) is the moment sequence representation. - Paul Barry, Jan 11 2007

Row sums of triangle A132812 starting (1, 4, 15, 56, 210, ...). - Gary W. Adamson, Sep 01 2007

Starting (1, 4, 15, 56, 210, ...) gives the binomial transform of A025566 starting (1, 3, 8, 22, 61, 171, ...). - Gary W. Adamson, Sep 01 2007

For n >= 1, a(2^n) = 2^(n+1)*A001795(2^(n-1)). - Vladimir Shevelev, Sep 05 2010

D-finite with recurrence: (n-1)*(n+1)*a(n) = 2*n*(2n-1)*a(n-1). - R. J. Mathar, Dec 17 2011

From Sergei N. Gladkovskii, Jul 07 2012: (Start)

G.f.: -1/(2*x) - G(0) where G(k) = 1 - 1/(2*x - 8*x^3*(2*k+1)/(4*x^2*(2*k+1)- (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step);

E.g.f.: BesselI(1,2*x)*exp(2*x) = x*G(0) where G(k) = 1 + 2*x*(4*k+3)/((2*k+1)*(2*k+3) - x*(2*k+1)*(2*k+3)*(4*k+5)/(x*(4*k+5) + 2*(k+1)*(k+2)/G(k+1))); (continued fraction, 3rd kind, 3-step).

(End)

G.f.: c(x)^3/(2-c(x)) where c(x) is the g.f. for A000108. - Cheyne Homberger, May 05 2014

G.f.: z*C(z)^2/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015

G.f.: x*2F1(3/2,2;3;4x). - R. J. Mathar, Aug 09 2015

a(n) = Sum_{i=1..n} binomial(2*i-2,i-1)*binomial(2*(n-i+1),n-i+2)/(n-i+1). - Vladimir Kruchinin, Sep 07 2015

L.g.f.: 1/(1 - x/(1 - x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Sum_{n>=1} 1/a(n) = 1/3 + 5*Pi/(9*sqrt(3)). - Amiram Eldar, Dec 04 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/5 + 14*sqrt(5)*log(phi)/25, where log(phi) = A002390. - Amiram Eldar, Feb 20 2021

a(n) = Product_{i=1..(n - 1)} (((4*i + 6)*i + 2)/((i + 2)*i)), for n>=1. - Neven Sajko, Oct 10 2021

MATHEMATICA

Table[Binomial[2n, n-1], {n, 0, 30}] (* Harvey P. Dale, Jul 12 2012 *)

CoefficientList[ Series[(1 - 2x - Sqrt[1 - 4x])/(2x*Sqrt[1 - 4x]), {x, 0, 26}], x] (* Robert G. Wilson v, Aug 10 2018 *)

PROG

(PARI) a(n)=if(n<1, 0, (2*n)!/(n+1)!/(n-1)!)

(Maxima) A001791(n):=binomial(2*n, n-1)$

makelist(A001791(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

(MAGMA) [Binomial(2*n, n-1): n in [0..30]]; // Vincenzo Librandi, Apr 20 2015

(GAP) List([0..30], n->Binomial(2*n, n-1)); # Muniru A Asiru, Aug 09 2018

CROSSREFS

Diagonal 3 of triangle A100257.

First differences are in A076540.

Cf. A000108, A000984, A002378, A002390, A025566, A132812.

A345197 counts compositions by length and alternating sum.

Cf. A000070, A000302, A000346, A002054, A008549, A032443, A088218, A097805, A163493, A202736, A345910.

Sequence in context: A026030 A047038 A158500 * A047128 A087438 A131497

Adjacent sequences:  A001788 A001789 A001790 * A001792 A001793 A001794

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)