A003409 -id:A003409 - cp's OEIS frontend

A029651 Central elements of the (1,2)-Pascal triangle A029635.

1, 3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650

Offset: 0

Mohammad K. Azarian

If Y is a fixed 2-subset of a (2n+1)-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

V. N. Smith and L. Shapiro, Catalan numbers, Pascal's triangle and mutators, Congressus Numerant., 205 (2010), 187-197.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
C. Bean, M. Tannock and H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015.
Milan Janjic, Two Enumerative Functions
Mark C. Wilson, Asymptotics for generalized Riordan arrays, International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005. (However, the asymptotics given there on p. 328 for a(n) give different results for me. - _Ralf Stephan_, Dec 28 2013)

Cf. A001700, A143398, A145460.

Essentially a duplicate of A003409.

a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n))-0^n/2;
seq(simplify(a(n)), n=0..24); # Peter Luschny, Dec 16 2015

Join[{1},Table[3*Binomial[2n-1,n],{n,30}]] (* Harvey P. Dale, Aug 11 2015 *)

concat([1], for(n=1, 50, print1(3*binomial(2*n-1,n), ", "))) \\ G. C. Greubel, Jan 23 2017

a(n) = 3 * binomial(2n-1, n) (n>0). - Len Smiley, Nov 03 2001
a(n) = 3*A001700(n-1), (n>=1).
G.f.: (1+xC(x))/(1-2xC(x)), C(x) the g.f. of A000108. - Paul Barry, Dec 17 2004
a(n) = A003409(n), n>0. - R. J. Mathar, Oct 23 2008
a(n) = Sum_{k=0..n} A039599(n,k)*A000034(k). - Philippe Deléham, Oct 29 2008
a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n))-0^n/2. - Peter Luschny, Dec 16 2015
a(n) ~ (3/2)*4^n*(1-(1/8)/n+(1/128)/n^2+(5/1024)/n^3-(21/32768)/n^4)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
a(n) = 2^(1-n)*Sum_{k=0..n} binomial(k+n,k)*binomial(2*n-1,n-k), n>0, a(0)=1. - Vladimir Kruchinin, Nov 23 2016
E.g.f.: (3*exp(2*x)*BesselI(0,2*x) - 1)/2. - Ilya Gutkovskiy, Nov 23 2016
a(n) = A143398(2n,n) = A145460(2n,n). - Alois P. Heinz, Sep 09 2018
a(n) = [x^n] C(-x)^(-3*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Oct 16 2024

More terms from David W. Wilson

A200777 T(n,k)=Number of nXk 0..2 arrays with every row and column running average nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

Original entry on oeis.org

3, 3, 3, 2, 5, 2, 7, 5, 5, 7, 9, 20, 13, 20, 9, 6, 37, 31, 31, 37, 6, 22, 31, 77, 342, 77, 31, 22, 30, 144, 217, 1368, 1368, 217, 144, 30, 20, 273, 553, 1806, 7725, 1806, 553, 273, 20, 75, 218, 1438, 18801, 13422, 13422, 18801, 1438, 218, 75, 105, 1054, 3849, 66784, 194535

Offset: 1

R. H. Hardin Nov 22 2011

Table starts
..3....3....2......7........9.........6..........22...........30............20
..3....5....5.....20.......37........31.........144..........273...........218
..2....5...13.....31.......77.......217.........553.........1438..........3849
..7...20...31....342.....1368......1806.......18801........66784.........80772
..9...37...77...1368.....7725.....13422......194535.......980702.......1678693
..6...31..217...1806....13422.....91755......633413......4472626......30905995
.22..144..553..18801...194535....633413....17965438....171117355.....537341613
.30..273.1438..66784...980702...4472626...171117355...2189118202....9237040535
.20..218.3849..80772..1678693..30905995...537341613...9237040535..159001066390
.75.1054.9634.855668.24878620.209517906.14614583388.342978784324.2722231659577

			Some solutions for n=6 k=4
..0..0..0..0....0..0..1..1....0..0..0..0....0..0..0..0....0..0..0..0
..0..1..2..1....0..0..1..2....0..1..2..1....0..0..2..2....0..0..2..1
..0..2..1..2....0..0..1..2....0..1..2..1....0..0..1..1....0..1..2..1
..0..1..1..2....0..1..1..2....0..1..2..1....1..1..1..2....0..1..2..2
..0..1..2..2....0..2..1..2....0..2..2..2....1..1..2..2....1..1..2..2
..1..1..2..2....1..2..2..2....1..1..2..2....1..2..2..2....1..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..178

Column 1 includes A003409((n+1)/3) and A000984(n/3)

A200770 Number of nX1 0..2 arrays with every row and column running average nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

Original entry on oeis.org

3, 3, 2, 7, 9, 6, 22, 30, 20, 75, 105, 70, 266, 378, 252, 966, 1386, 924, 3564, 5148, 3432, 13299, 19305, 12870, 50050, 72930, 48620, 189618, 277134, 184756, 722228, 1058148, 705432, 2762942, 4056234, 2704156, 10608612, 15600900, 10400600, 40859500

Offset: 1

R. H. Hardin Nov 22 2011

nonn

Column 1 of A200777

			All solutions for n=4
..0....0....0....0....0....0....0
..0....2....0....1....1....1....2
..1....1....2....2....2....1....1
..2....2....1....2....1....2....1

R. H. Hardin, Table of n, a(n) for n = 1..210

Includes A003409((n+1)/3) and A000984(n/3)

Conjecture: (n+2)*(2270*n-4893)*a(n) +3*(1090*n^2-4919*n+1654)*a(n-1) +4686*a(n-2) +2*(-5540*n^2+7016*n+17127)*a(n-3) -6*(1090*n-2739)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 23 2011

A370487 Number of partitions of [3n] into 3 sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

Original entry on oeis.org

1, 1, 7, 28, 103, 376, 1384, 5146, 19303, 72928, 277132, 1058146, 4056232, 15600898, 60174898, 232676278, 901620583, 3500409328, 13612702948, 53017895698, 206769793228, 807386811658, 3156148445578, 12350146091398, 48371405524648, 189615909656626, 743877799422154

Offset: 0

Alois P. Heinz, Feb 19 2024

nonn

			a(0) = 1: {}|{}|{}.
a(1) = 1: 1|2|3.
a(2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.

Alois P. Heinz, Table of n, a(n) for n = 0..1663
Wikipedia, Partition of a set

Row n=3 of A370363.

Cf. A000079, A001700, A003409, A029651, A131577, A255738.

a:= n-> `if`(n=0, 1, 3*binomial(2*n-1,n)-2):
seq(a(n), n=0..27);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 3*n*(n-1)+1, ((15*n^2
      -31*n+12)*a(n-1)-(3*n-2)*(4*n-6)*a(n-2))/((3*n-5)*n))
    end:
seq(a(n), n=0..27);

a(n) = 3*binomial(2*n-1,n) - 2 for n >= 1, a(0) = 1.
G.f.: ((3*x+1)*(4*x-1)+3*(1-x)*sqrt(1-4*x))/(2*(1-x)*(1-4*x)).
a(n) = A003409(n) - 2 = A029651(n) - 2 = 3*A001700(n-1) - 2 for n >= 1.
a(n) mod 2 = A255738(n+1).
a(n) mod 2 = 1 <=> n in { A131577 }.

A334456 Number h of points and of blocks of nontrivial biplanes.

Original entry on oeis.org

7, 11, 16, 37, 56, 79, 121

Offset: 1

Stefano Spezia, Apr 30 2020

A biplane is an incidence structure consisting of a set P of h points and a set of h blocks, each of which is a k-subset of P (block size k), such that every pair of points lies in exactly 2 blocks (see Alavi et al.).

Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See Introduction p. 1.
P. Kaski and P. R. J. Östergård. There are exactly five biplanes with k = 11, J. Combin. Des., 16(2):117-127, 2008.

Cf. A001109, A003409, A038723, A038725, A054490, A334457 (k).

A334457 Block sizes k of nontrivial biplanes.

Original entry on oeis.org

4, 5, 6, 9, 11, 13, 16

Offset: 1

Stefano Spezia, Apr 30 2020

A biplane is an incidence structure consisting of a set P of h points and a set of h blocks, each of which is an k-subset of P (block size k), such that every pair of points lies in exactly 2 blocks (see Alavi et al.).

Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger, Symmetries of biplanes, arXiv:2004.04535 [math.GR], 2020. See Introduction p. 1.
P. Kaski and P. R. J. Östergård. There are exactly five biplanes with k = 11, J. Combin. Des., 16(2):117-127, 2008.

Cf. A001109, A003409, A038723, A038725, A054490, A334456 (h).

A128727 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DDU and LDU's.

Original entry on oeis.org

1, 1, 3, 9, 1, 27, 9, 81, 54, 2, 243, 270, 30, 729, 1215, 270, 5, 2187, 5103, 1890, 105, 6561, 20412, 11340, 1260, 14, 19683, 78732, 61236, 11340, 378, 59049, 295245, 306180, 85050, 5670, 42, 177147, 1082565, 1443420, 561330, 62370, 1386, 531441

Offset: 0

Emeric Deutsch, Mar 31 2007

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of steps in it.
Row n has ceiling(n/2) terms (n >= 1).
Row sums yield A002212.
Apparently a(n) = A126177(n-1). - Georg Fischer, Oct 28 2018

			T(5,2)=2 because we have UU(DDU)U(DDU)D and UUU(DDU)(DDU)D (the 2 subwords are shown between parentheses).
Triangle starts:
    1;
    1;
    3;
    9,    1;
   27,    9;
   81,   54,    2;
  243,  270,   30;
  729, 1215,  270,    5;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A000108, A002212, A003409, A026377, A126177.

T:=(n,k)->3^(n-1-2*k)*binomial(n,k)*binomial(n-k,k+1)/n: 1; for n from 1 to 13 do seq(T(n,k),k=0..floor((n-1)/2)) od; # yields sequence in triangular form

T(n,0) = 3^(n-1).
T(2k+1,k) = binomial(2k,k)/(k+1) (the Catalan numbers, A000108).
T(2k,k-1) = 3binomial(2k-1,k) = A003409(k).
Sum_{k>=0} k*T(n,k) = A026377(n-1).
T(n,k) = (1/n)*3^(n-1-2k)*binomial(n,k)*binomial(n-k,k+1).
G.f.: G = G(t,z) satisfies tzG^2 - (1 - 3z + 2tz)G + 1 - 2z + tz = 0.

cp's OEIS Frontend

A029651 Central elements of the (1,2)-Pascal triangle A029635.

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A200777 T(n,k)=Number of nXk 0..2 arrays with every row and column running average nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

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A200770 Number of nX1 0..2 arrays with every row and column running average nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

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Author

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Comments

Examples

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Crossrefs

Formula

A370487 Number of partitions of [3n] into 3 sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A334456 Number h of points and of blocks of nontrivial biplanes.

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Author

Keywords

Comments

Links

Crossrefs

A334457 Block sizes k of nontrivial biplanes.

Views

Author

Keywords

Comments

Links

Crossrefs

A128727 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DDU and LDU's.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula