A008549 -id:A008549 - cp's OEIS frontend

A129182 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2).

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 7, 0, 7, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 14, 0, 17, 0, 16, 0, 16, 0, 14, 0, 11, 0, 9, 0, 7, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 0

Emeric Deutsch, Apr 08 2007

Row n has n^2 + 1 terms.
Row sums are the Catalan numbers (A000108).
Sum(k*T(n,k), k=0..n^2) = A008549(n).
Sums along falling diagonals give A005169. - Joerg Arndt, Mar 29 2014
T(2n,4n) = A240008(n). - Alois P. Heinz, Mar 30 2014

			T(4,10) = 3 because we have UDUUUDDD, UUUDDDUD and UUDUDUDD.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,2,0,1,0,1;
0,0,0,0,1,0,3,0,3,0,3,0,2,0,1,0,1;
0,0,0,0,0,1,0,4,0,6,0,7,0,7,0,5,0,5,0,3,0,2,0,1,0,1;
Transposed triangle (A239927) begins:
00:  1;
01:  0, 1;
02:  0, 0, 1;
03:  0, 0, 0, 1;
04:  0, 0, 1, 0, 1;
05:  0, 0, 0, 2, 0, 1;
06:  0, 0, 0, 0, 3, 0, 1;
07:  0, 0, 0, 1, 0, 4, 0, 1;
08:  0, 0, 0, 0, 3, 0, 5, 0, 1;
09:  0, 0, 0, 1, 0, 6, 0, 6, 0, 1;
10:  0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;
11:  0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;
12:  0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;
13:  0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;
14:  0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;
15:  0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;
16:  0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;
17:  0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;
18:  0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;
19:  0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;
20:  0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;
... - _Joerg Arndt_, Mar 25 2014

Alois P. Heinz, Rows n = 0..32, flattened

Cf. A000108, A008549, A139262, A240008, A143951 (column sums).

G:=1/(1-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G,z=0,11)): for n from 0 to 7 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
       expand(b(x-1, y-1)*z^(y-1/2)+ b(x-1, y+1)*z^(y+1/2))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 29 2014

b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2) + b[x-1, y+1]*z^(y+1/2)]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

G.f.: G(t,z) given by G(t,z) = 1+t*z*G(t,t^2*z)*G(t,z).

Sum_{k=0..n^2} (n^2-k)/2 * T(n,k) = A139262(n). - Alois P. Heinz, Mar 31 2018

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.

Original entry on oeis.org

0, 1, 9, 87, 975, 12645, 187425, 3133935, 58437855, 1203216525, 27125492625, 664761133575, 17600023616175, 500706514833525, 15234653491682625, 493699195087473375, 16977671416936605375, 617528830880480644125, 23687738668934964248625

Offset: 1

Wolfdieter Lang

2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n-1)+ A001147(n-1), n >= 2, a(1)=0.
a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n non-root vertices. See the example a(3)=9 given in A035342. Organic labeling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. - Wolfdieter Lang, Aug 07 2007
a(n), n>=2, enumerates unordered n-vertex forests composed of two plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly 1 of the remaining n-1 chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020

			a(2)=1 for the forest: {r1-1, r2-2} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).
a(3)=9 increasing ternary 2-forest with n=3 vertices: there are three 2-forests (the one vertex tree together with any of the three different 2-vertex trees) each with three increasing labelings. - _Wolfdieter Lang_, Sep 14 2007

Robert Israel, Table of n, a(n) for n = 1..370
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A000108, A002866, A008549, A336599.

Cf. A001147 (m=1 column of A035342). See a D. Callan comment there on the number of increasing ordered rooted trees on n+1 vertices.

I:=[0,1,9]; [n le 3 select I[n] else - 2*(n-1)*(2*n-3)*Self(n-2)+(4*n-3)*Self(n-1): n in [1..30]]; // Vincenzo Librandi, Sep 12 2015

F:= gfun:-rectoproc({(4*n^2+6*n+2)*a(n)+(-4*n-5)*a(n+1)+a(n+2),a(1)=0,a(2)=1,a(3)=9},a(n),remember):
map(f, [$1..30]); # Robert Israel, Sep 11 2015

Table[Round [n! (4^(n - 1) - Binomial[2 n, n]/2)/2^(n - 1)], {n, 1, 20}] (* Vincenzo Librandi, Sep 12 2015 *)

a(n) = n!*(4^(n-1)-binomial(2*n, n)/2)/2^(n-1);
vector(40, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = n!*A008549(n-1)/2^(n-1) = n!(4^(n-1)-binomial(2*n, n)/2)/2^(n-1).
a(n) = (2n-2)*a(n-1) + A129890(n-2). - Philippe Deléham, Oct 28 2013
a(n) = n!*2^(n-1) - A001147(n) = A002866(n) - A001147(n). - Peter Bala, Sep 11 2015
a(n) = -2*(n-1)*(2*n-3)*a(n-2)+(4*n-3)*a(n-1). - Robert Israel, Sep 11 2015

A045720 3-fold convolution of A001700(n), n >= 0.

Original entry on oeis.org

1, 9, 57, 312, 1578, 7599, 35401, 161052, 719790, 3173090, 13836426, 59803104, 256596276, 1094249019, 4642178601, 19605872724, 82483419846, 345839048094, 1445715336366, 6027524015664, 25070662980876, 104056307673654

Offset: 0

Wolfdieter Lang

Total number of 132 (or 213) patterns in the set of all 123-avoiding permutations of length (n+3). - Cheyne Homberger, Mar 16 2012

a(n) is the degree of the cyclic graphical Gaussian model for the (n+3) cycle. - Mateusz Michalek, Mar 04 2023

B. Sturmfels, and C. Uhler. Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry. Annals of the Institute of Statistical Mathematics 62.4 (2010): 603-638, Conjecture 2 proved in "Geometry of the Gaussian graphical model of the cycle"

Indranil Ghosh, Table of n, a(n) for n = 0..1500
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.
A. Ayyer, Towards a Human Proof of Gessel's Conjecture, JIS 12 (2009) 09.4.2
R. Dinu, M. Michalek, and M. Vodička. Geometry of the Gaussian graphical model of the cycle, arXiv preprint arXiv:2111.02937 [math.AG] (2021).
C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
D. R. Snow, Spreadsheets, Power Series, Generating Functions and Integers, The College Maths. J. 20 (1989) 149.

Cf. A000108, A001700, A008549.

Table[(n+5)*Binomial[2*(n+3),n+3]/4-3*2^(2n+3),{n,0,21}] (* Indranil Ghosh, Feb 18 2017 *)

x='x+O('x^30); Vec((((1-4*x)^(-1/2)-1)/(2*x))^3) \\ Altug Alkan, Sep 04 2018

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A045720(n):
    return (n+5)*C(2*(n+3),n+3)/4-3*2**(2*n+3) # Indranil Ghosh, Feb 18 2017

a(n) = (n+5)*binomial(2*(n+3), n+3)/4 - 3*2^(2*n+3);
G.f.: (c(x)/sqrt(1-4*x))^3, where c(x) = g.f. for Catalan numbers A000108;
recursion: a(n)=(2*(2*n+7)/(n+3))*a(n-1)+(3/(n+3))*A008549(n+1), a(0)=1.

A345926 Number of distinct possible alternating sums of permutations of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3

Offset: 1

Gus Wiseman, Jul 14 2021

nonn

First differs from A096825 at a(90) = 3, A096825(90) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also the number of possible values of A056239(d) where d is a divisor of n with half as many prime factors (rounded up) as n.

			Grouping the 12 permutations of {1,2,2,3} by alternating sum k gives:
  k = -2: (1223) (1322) (2213) (2312)
  k =  0: (1232) (2123) (2321) (3212)
  k =  2: (2132) (2231) (3122) (3221)
so a(90) = 3.

The version for prime factors instead of indices is A343943.
A000005 counts divisors.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices,  row sums of A112798.
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).
A345197 counts compositions by length and alternating sum.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A008549, A032443, A083399, A096825, A239830, A344607, A344609, A344651, A345957, A345960, A345961.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Union[ats/@Permutations[primeMS[n]]]],{n,100}]

from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A345926(n):
    fs = dict((primepi(a),b) for (a,b) in factorint(n).items())
    return len(set(sum(d) for d in multiset_combinations(fs, (sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).

Original entry on oeis.org

1, 2, 5, 1, 14, 6, 42, 27, 1, 132, 110, 10, 429, 429, 65, 1, 1430, 1638, 350, 14, 4862, 6188, 1700, 119, 1, 16796, 23256, 7752, 798, 18, 58786, 87210, 33915, 4655, 189, 1, 208012, 326876, 144210, 24794, 1518, 22, 742900, 1225785, 600875, 123970, 10350

Offset: 0

Emeric Deutsch, May 06 2006

Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.
Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna, May 10 2006
Column k is the sum of columns 2k and 2k+1 of A039599. - Philippe Deléham, Nov 11 2008

			T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \).
Triangle starts:
  1;
  2;
  5,1;
  14,6;
  42,27,1;
  132,110,10;

Cf. A000984, A000108, A003517, A003519, A008549, A118920.

Cf. A119243 (eigenvector), A119245 (variant).

T:=(n,k)->(2*k+1)*binomial(2*n+2,n-2*k)/(n+1): for n from 0 to 13 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

T(n,k)=if(n<2*k || k<0,0,(2*k+1)*binomial(2*n+2,n-2*k)/(n+1)) \\ Paul D. Hanna, May 10 2006

T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

T(n,k)=A039599(n,2k)+A039599(n,2k+1). - Philippe Deléham, Nov 11 2008

A345907 Triangle giving the main antidiagonals of the matrices counting integer compositions by length and alternating sum (A345197).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 4, 3, 1, 1, 0, 0, 3, 6, 4, 1, 1, 0, 0, 6, 9, 8, 5, 1, 1, 0, 0, 0, 18, 18, 10, 6, 1, 1, 0, 0, 0, 10, 36, 30, 12, 7, 1, 1, 0, 0, 0, 20, 40, 60, 45, 14, 8, 1, 1, 0, 0, 0, 0, 80, 100, 90, 63, 16, 9, 1, 1

Offset: 0

Gus Wiseman, Jul 26 2021

The matrices (A345197) count the integer compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2. Here, the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Problem: What are the column sums? They appear to match A239201, but it is not clear why.

			Triangle begins:
   1
   1   1
   0   1   1
   0   1   1   1
   0   2   2   1   1
   0   0   4   3   1   1
   0   0   3   6   4   1   1
   0   0   6   9   8   5   1   1
   0   0   0  18  18  10   6   1   1
   0   0   0  10  36  30  12   7   1   1
   0   0   0  20  40  60  45  14   8   1   1
   0   0   0   0  80 100  90  63  16   9   1   1
   0   0   0   0  35 200 200 126  84  18  10   1   1
   0   0   0   0  70 175 400 350 168 108  20  11   1   1
   0   0   0   0   0 350 525 700 560 216 135  22  12   1   1

Row sums are A163493.
Rows are the antidiagonals of the matrices given by A345197.
The main diagonals of A345197 are A346632, with sums A345908.
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).
Other diagonals are A008277 of A318393 and A055884 of A320808.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A007318, A008549, A025047, A114121, A344610.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{n-k}],k==(n+ats[#])/2-1&]],{k,0,n-1}],{n,0,15}]

A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

Original entry on oeis.org

0, 9, 130, 135, 141, 153, 177, 193, 225, 2052, 2059, 2062, 2069, 2074, 2079, 2089, 2098, 2103, 2109, 2129, 2146, 2151, 2157, 2169, 2209, 2242, 2247, 2253, 2265, 2289, 2369, 2434, 2439, 2445, 2457, 2481, 2529, 2561, 2689, 2818, 2823, 2829, 2841, 2865, 2913

Offset: 1

Gus Wiseman, Nov 22 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
     0: ()
     9: (3,1)
   130: (6,2)
   135: (5,1,1,1)
   141: (4,1,2,1)
   153: (3,1,3,1)
   177: (2,1,4,1)
   193: (1,6,1)
   225: (1,1,5,1)
  2052: (9,3)
  2059: (8,2,1,1)
  2062: (8,1,1,2)
  2069: (7,2,2,1)
  2074: (7,1,2,2)
  2079: (7,1,1,1,1,1)
  2089: (6,2,3,1)
  2098: (6,1,3,2)
  2103: (6,1,2,1,1,1)

These compositions are counted by A224274 up to 0's.
An unordered version is A348617, counted by A001523 up to 0's.
The positive version is A349153, unreversed A348614.
The unreversed version is A349154.
Positive unordered unreversed: A349159, counted by A000712 up to 0's.
A positive unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==-2*sats[stc[#]]&]

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 12, 126, 1268, 12513, 122148, 1184364, 11432100, 109997460, 1055891248, 10117633542, 96812495820, 925334377822, 8836315646616, 84317468847768, 804064275489924, 7663595943744876, 73009005101019792, 695263276434909976, 6618709687608909648, 62989317586872238689

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A308523, A386565, A386612.

Cf. A008549, A386613, A386615.

a(n) = sum(k=0, n-1, binomial(4*k, k)*binomial(4*n-4*k, n-k-1));

G.f.: g^2 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g) * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+1,k).

A386613 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k,n-k-1).

Original entry on oeis.org

0, 1, 15, 200, 2570, 32470, 406411, 5057440, 62692100, 775007135, 9561421830, 117780193480, 1449107627450, 17811990468400, 218768774024360, 2685209277718320, 32940971570389960, 403920568087927025, 4950915045235523125, 60663591616305306320, 743092566613017730980, 9100088494955802407060

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A079678, A386367, A386566, A386614.

Cf. A008549, A386611, A386615.

a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k, n-k-1));

G.f.: g^2 * (g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/((1-g) * (1-5*g)^2) where g*(1-g)^4 = x.
a(n) = Sum_{k=0..n-1} binomial(5*k+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k+1,k).

A386615 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k,n-k-1).

Original entry on oeis.org

0, 1, 18, 291, 4550, 70065, 1069872, 16251694, 246010014, 3714826350, 55993450830, 842823848448, 12672667549488, 190381643518855, 2858101359683400, 42882348756992220, 643085584745669134, 9640075656634321770, 144457232389535563980, 2164044325920832653825, 32409930873969839549610

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A079679, A386368, A386567, A386616.

Cf. A008549, A386611, A386613.

a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k, n-k-1));

G.f.: g^2 * (g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/((1-g) * (1-6*g)^2) where g*(1-g)^5 = x.
a(n) = Sum_{k=0..n-1} binomial(6*k+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n+1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k+1,k).

cp's OEIS Frontend

A129182 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2).

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A035101 E.g.f. x*(c(x/2)-1)/(1-2*x), where c(x) = g.f. for Catalan numbers A000108.

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A045720 3-fold convolution of A001700(n), n >= 0.

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A345926 Number of distinct possible alternating sums of permutations of the multiset of prime indices of n.

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A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).

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A345907 Triangle giving the main antidiagonals of the matrices counting integer compositions by length and alternating sum (A345197).

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A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

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A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

A386613 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k,n-k-1).

A386615 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k,n-k-1).