A045894 -id:A045894 - cp's OEIS frontend

A008549 Number of ways of choosing at most n-1 items from a set of size 2*n+1.

0, 1, 6, 29, 130, 562, 2380, 9949, 41226, 169766, 695860, 2842226, 11576916, 47050564, 190876696, 773201629, 3128164186, 12642301534, 51046844836, 205954642534, 830382690556, 3345997029244, 13475470680616, 54244942336114, 218269673491780, 877940640368572

Offset: 0

N. J. A. Sloane

Area under Dyck excursions (paths ending in 0): a(n) is the sum of the areas under all Dyck excursions of length 2*n (nonnegative walks beginning and ending in 0 with jumps -1,+1).
Number of inversions in all 321-avoiding permutations of [n+1]. Example: a(2)=6 because the 321-avoiding permutations of [3], namely 123,132,312,213,231, have 0, 1, 2, 1, 2 inversions, respectively. - Emeric Deutsch, Jul 28 2003
Convolution of A001791 and A000984. - Paul Barry, Feb 16 2005
a(n) = total semilength of "longest Dyck subpath" starting at an upstep U taken over all upsteps in all Dyck paths of semilength n. - David Callan, Jul 25 2008
[1,6,29,130,562,2380,...] is convolution of A001700 with itself. - Philippe Deléham, May 19 2009
From Ran Pan, Feb 04 2016: (Start)
a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) bounce off the diagonal y = x to the right. This is related to paired pattern P_2 in Pan and Remmel's link and more details can be found in Section 3.2 in the link.
a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) horizontally cross the diagonal y = x. This is related to paired pattern P_3 in Pan and Remmel's link and more details can be found in Section 3.3 in the link.
2*a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) bounce off the diagonal y = x. This is related to paired pattern P_2 and P_4 in Pan and Remmel's link and more details can be found in Section 4.2 in the link.
2*a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) cross the diagonal y = x. This is related to paired pattern P_3 and P_4 in Pan and Remmel's link and more details can be found in Section 4.3 in the link. (End)
From Gus Wiseman, Jul 17 2021: (Start)
Also the number of integer compositions of 2*(n+1) with alternating sum < 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 29 compositions of 8 are:
  (1,7)  (1,5,2)  (1,1,1,5)  (1,1,1,4,1)  (1,1,1,1,1,3)
  (2,6)  (1,6,1)  (1,1,2,4)  (1,2,1,3,1)  (1,1,1,2,1,2)
  (3,5)  (2,5,1)  (1,2,1,4)  (1,3,1,2,1)  (1,1,1,3,1,1)
                  (1,2,2,3)  (1,4,1,1,1)  (1,2,1,1,1,2)
                  (1,3,1,3)               (1,2,1,2,1,1)
                  (1,3,2,2)               (1,3,1,1,1,1)
                  (1,4,1,2)
                  (1,4,2,1)
                  (1,5,1,1)
                  (2,1,1,4)
                  (2,2,1,3)
                  (2,3,1,2)
                  (2,4,1,1)
Also the number of integer compositions of 2*(n+1) with reverse-alternating sum < 0. For a bijection, keep the odd-length compositions and reverse the even-length ones.
Also the number of 2*(n+1)-digit binary numbers with more 0's than 1's. For example, the a(2) = 6 binary numbers are: 100000, 100001, 100010, 100100, 101000, 110000; or in decimal: 32, 33, 34, 36, 40, 48.
(End)

			a(2) = 6 because there are 6 ways to choose at most 1 item from a set of size 5: You can choose the empty set, or you can choose any of the five one-element sets.
G.f. = x + 6*x^2 + 29*x^3 + 130*x^4 + 562*x^5 + 2380*x^6 + 9949*x^7 + ...

D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.

Indranil Ghosh, Table of n, a(n) for n = 0..1500 (terms 0..200 from T. D. Noe)
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, 18 (2015), Article 15.5.1.
Octavio Arizmendi, Daniel Perales, and Josue Vazquez-Becerra, Finite Free Convolution: Infinitesimal Distributions, arXiv:c [math.PR], 2025. See p. 34.
Jean Christophe Aval, Adrien Boussicault, Patxi Laborde-Zubieta, and Mathias Pétréolle, Generating series of Periodic Parallelogram polyominoes, arXiv:1612.03759, 2016.
Roland Bacher, On generating series of complementary plane trees, arXiv:math/0409050 [math.CO], 2004.
Vijay Balasubramanian, Javier M. Magan, and Qingyue Wu, A Tale of Two Hungarians: Tridiagonalizing Random Matrices, arXiv:2208.08452 [hep-th], 2022.
Cyril Banderier, Analytic combinatorics of random walks and planar maps, Ph. D. Thesis, 2001. [Broken link]
Adrien Boussicault and P. Laborde-Zubieta, Periodic Parallelogram Polyominoes, arXiv preprint arXiv:1611.03766 [math.CO], 2016.
AJ Bu, Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers), arXiv:2310.17026 [math.CO], 2023.
AJ Bu and Doron Zeilberger, Using Symbolic Computation to Explore Generalized Dyck Paths and Their Areas, arXiv:2305.09030 [math.CO], 2023.
Alexander Burstein and Sergi Elizalde, Total occurrence statistics on restricted permutations, arXiv:1305.3177 [math.CO], 2013.
Robin Chapman, Moments of Dyck paths, Discrete Math., 204 (1999), 113-117.
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Niklas G. Johansson, Efficient Simulation of the Deutsch-Jozsa Algorithm, Master's Project, Department of Electrical Engineering & Department of Physics, Chemistry and Biology, Linkoping University, April, 2015.
Miles Jones, Sergey Kitaev, and Jeffrey Remmel, Frame patterns in n-cycles, arXiv preprint arXiv:1311.3332 [math.CO], 2013.
James A. Mingo and Josue Vazquez-Becerra, The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix, arXiv:2112.15231 [math.PR], 2021.
Henri Mühle, Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices, arXiv:1509.06942v1 [math.CO], 2015.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Elisa Pergola, Two bijections for the area of Dyck paths, Discrete Math., 241 (2001), 435-447.
Wen-jin Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.

Cf. A038608, A045894, A057571, A109196, A153338.
Odd bisection of A294175 (even is A000346).
For integer compositions of 2*(n+1) with alternating sum k < 0 we have:
- The opposite (k > 0) version is A000302.
- The weak (k <= 0) version is (also) A000302.
- The k = 0 version is A001700 or A088218.
- The reverse-alternating version is also A008549 (this sequence).
- These compositions are ranked by A053754 /\ A345919.
- The complement (k >= 0) is counted by A114121.
- The case of reversed integer partitions is A344743(n+1).
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A345197 counts compositions by length and alternating sum.
Cf. A000070, A001791, A007318, A025047, A027306, A032443, A058622, A120452, A163493, A239830, A344611.

[4^n-Binomial(2*n+1, n): n in [0..30]]; // Vincenzo Librandi, Feb 04 2016

A008549:=n->4^n-binomial(2*n+1,n): seq(A008549(n), n=0..30);

Table[4^n-Binomial[2n+1,n],{n,0,30}] (* Harvey P. Dale, May 11 2011 *)
a[ n_] := If[ n<-4, 0, 4^n - Binomial[2 n + 2, n + 1] / 2] (* Michael Somos, Jan 25 2014 *)

{a(n)=if(n<0, 0, 4^n - binomial(2*n+1, n))} /* Michael Somos Oct 31 2006 */

{a(n) = if( n<-4, 0, n++; (4^n / 2 - binomial(2*n, n)) / 2)} /* Michael Somos, Jan 25 2014 */

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A008549(n):
    return int((4**n)-C(2*n+1,n)) # Indranil Ghosh, Feb 18 2017

a(n) = 4^n - C(2*n+1, n).
a(n) = Sum_{k=1..n} Catalan(k)*4^(n-k): convolution of Catalan numbers and powers of 4.
G.f.: x*c(x)^2/(1 - 4*x), c(x) = g.f. of Catalan numbers. - Wolfdieter Lang
Note Sum_{k=0..2*n+1} binomial(2*n+1, k) = 2^(2n+1). Therefore, by the symmetry of Pascal's triangle, Sum_{k=0..n} binomial(2*n+1, k) = 2^(2*n) = 4^n. This explains why the following two expressions for a(n) are equal: Sum_{k=0..n-1} binomial(2*n+1, k) = 4^n - binomial(2*n+1, n). - Dan Velleman
G.f.: (2*x^2 - 1 + sqrt(1 - 4*x^2))/(2*(1 + 2*x)*(2*x - 1)*x^3).
a(n) = Sum_{k=0..n} C(2*k, k)*C(2*(n-k), n-k-1). - Paul Barry, Feb 16 2005
Second binomial transform of 2^n - C(n, floor(n/2)) = A045621(n). - Paul Barry, Jan 13 2006
a(n) = Sum_{0 < i <= k < n} binomial(n, k+i)*binomial(n, k-i). - Mircea Merca, Apr 05 2012
D-finite with recurrence (n+1)*a(n) + 2*(-4*n-1)*a(n-1) + 8*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012
0 = a(n) * (256*a(n+1) - 224*a(n+2) + 40*a(n+3)) + a(n+1) * (-32*a(n+1) + 56*a(n+2) - 14*a(n+3)) + a(n+2) * (-2*a(n+2) + a(n+3)) if n > -5. - Michael Somos, Jan 25 2014
Convolution square is A045894. - Michael Somos, Jan 25 2014
HANKEL transform is [0, -1, 2, -3, 4, -5, ...]. - Michael Somos, Jan 25 2014
BINOMIAL transform of [0, 0, 1, 3, 11, 35,...] (A109196) is [0, 0, 1, 6, 29, 130, ...]. - Michael Somos, Jan 25 2014
(n+1) * a(n) = A153338(n+1). - Michael Somos, Jan 25 2014
a(n) = Sum_{m = n+2..2*n+1} binomial(2*n+1,m), n >= 0. - Wolfdieter Lang, May 22 2015
E.g.f.: (exp(2*x) - BesselI(0,2*x) - BesselI(1,2*x))*exp(2*x). - Ilya Gutkovskiy, Aug 30 2016

Better description from Dan Velleman (djvelleman(AT)amherst.edu), Dec 01 2000

A035324 A convolution triangle of numbers, generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 29, 9, 1, 126, 130, 57, 12, 1, 462, 562, 312, 94, 15, 1, 1716, 2380, 1578, 608, 140, 18, 1, 6435, 9949, 7599, 3525, 1045, 195, 21, 1, 24310, 41226, 35401, 19044, 6835, 1650, 259, 24, 1, 92378, 169766, 161052, 97954, 40963, 12021, 2450

Offset: 1

Wolfdieter Lang

Replacing each '2' in the recurrence by '1' produces Pascal's triangle A007318(n-1,m-1). The columns appear as A001700, A008549, A045720, A045894, A035330, ...
Triangle T(n,k), 1 <= k <= n, given by (0, 3/1, 1/3, 5/3, 3/5, 7/5, 5/7, 9/7, 7/9, 11/9, 9/11, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 28 2012
Riordan array (1, c(x)/sqrt(1-4x)) where c(x) = g.f. for Catalan numbers A000108, first column (k = 0) omitted. - Philippe Deléham, Jan 28 2012

			Triangle begins:
    1;
    3,   1;
   10,   6,   1;
   35,  29,   9,   1;
  126, 130,  57,  12,   1;
  462, 562, 312,  94,  15,   1;
Triangle (0, 3, 1/3, 5/3, 3/5, ...) DELTA (1,0,0,0,0,0, ...) has an additional first column (1,0,0,...).

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.

Cf. A000108, A007318, A039598.
Row sums: A049027(n), n >= 1.
Alternating row sums give A000108 (Catalan numbers).
If offset 0 (n >= m >= 0): convolution triangle based on A001700 (central binomial coeffs. of odd order).

a035324 n k = a035324_tabl !! (n-1) !! (k-1)
a035324_row n = a035324_tabl !! (n-1)
a035324_tabl = map snd $ iterate f (1, [1]) where
   f (i, xs)  = (i + 1, map (`div` (i + 1)) $
      zipWith (+) ((map (* 2) $ zipWith (*) [2 * i + 1 ..] xs) ++ [0])
                  ([0] ++ zipWith (*) [2 ..] xs))
-- Reinhard Zumkeller, Jun 30 2013

a[n_, m_] /; n >= m >= 1 := a[n, m] = 2*(2*(n-1) + m)*(a[n-1, m]/n) + m*(a[n-1, m-1]/n); a[n_, m_] /; n < m = 0; a[n_, 0] = 0; a[1, 1] = 1; Flatten[ Table[ a[n, m], {n, 1, 10}, {m, 1, n}]] (* Jean-François Alcover, Feb 21 2012, from first formula *)

@cached_function
def T(n, k):
    if n == 0: return n^k
    return sum(binomial(2*i-1, i)*T(n-1, k-i) for i in (1..k-n+1))
A035324 = lambda n,k: T(k, n)
for n in (1..8): print([A035324(n, k) for k in (1..n)]) # Peter Luschny, Aug 16 2016

a(n+1, m) = 2*(2*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n

G.f. for column m: ((x*c(x)/sqrt(1-4*x))^m)/x, where c(x) = g.f. for Catalan numbers A000108.

a(n, m) =: s2(3; n, m).

With offset 0 (0 <= k <= n), T(n,k) = Sum_{j>=0} A039598(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007

T(n+1,n) = 3*n = A008585(n).

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

T(n,m) = Sum_{k=m..n} k*binomial(k-1,k-m)*2^(k-m)*binomial(2*n-k-1,n-k)/n. - Vladimir Kruchinin, Aug 07 2013

A035330 5-fold convolution of A001700(n), n >= 0.

Original entry on oeis.org

1, 15, 140, 1045, 6835, 40963, 230720, 1240740, 6437890, 32468470, 160010280, 773624615, 3680728375, 17274086235, 80119845080, 367821324040, 1673528845710, 7554110698850, 33858536700040, 150802994850570

Offset: 0

Wolfdieter Lang

Fifth column of triangular array A035324.

Michael De Vlieger, Table of n, a(n) for n = 0..1650
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A000108, A045894, A035324.

Array[(#^2 + 27 # + 122) Binomial[2 (# + 5), # + 5]/24 - 5 (# + 8)*2^(2 # + 5) &, 20, 0] (* Michael De Vlieger, Sep 04 2018 *)

a(n) = (n^2+27*n+122)*binomial(2*(n+5), n+5)/24 - 5*(n+8)*2^(2*n+5) = A035324(n+5, 5);

G.f.: c(x)^5/(1-4*x)^(5/2), where c(x) = g.f. for Catalan numbers A000108.

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A035520 Fourth column of triangle A035342; related to A045894.

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A008549 Number of ways of choosing at most n-1 items from a set of size 2*n+1.

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A035324 A convolution triangle of numbers, generalizing Pascal's triangle A007318.

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A035330 5-fold convolution of A001700(n), n >= 0.

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