A014068 - OEIS

A014068

a(n) = binomial(n*(n+1)/2, n).

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004

Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..370

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Eric Weisstein's World of Mathematics, Graph Loop.

FORMULA

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001

For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004

a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005

a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009

From Peter Bala, Feb 27 2020: (Start)

a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).

Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)

a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024

EXAMPLE

From Gus Wiseman, Jan 25 2024: (Start)

The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):

{} {{1}} {{1},{2}} {{1},{2},{3}}

{{1},{1,2}} {{1},{2},{1,2}}

{{2},{1,2}} {{1},{2},{1,3}}

{{1},{2},{2,3}}

{{1},{3},{1,2}}

{{1},{3},{1,3}}

{{1},{3},{2,3}}

{{2},{3},{1,2}}

{{2},{3},{1,3}}

{{2},{3},{2,3}}

{{1},{1,2},{1,3}}

{{1},{1,2},{2,3}}

{{1},{1,3},{2,3}}

{{2},{1,2},{1,3}}

{{2},{1,2},{2,3}}

{{2},{1,3},{2,3}}

{{3},{1,2},{1,3}}

{{3},{1,2},{2,3}}

{{3},{1,3},{2,3}}

{{1,2},{1,3},{2,3}}

(End)

MATHEMATICA

Binomial[First[#], Last[#]]&/@With[{nn=20}, Thread[{Accumulate[ Range[ 0, nn]], Range[ 0, nn]}]] (* Harvey P. Dale, May 27 2014 *)

PROG

(Sage) [(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

(Magma) [Binomial(Binomial(n+1, 2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022

(Python)

from math import comb

def A014068(n): return comb(comb(n+1, 2), n) # Chai Wah Wu, Jul 14 2024

CROSSREFS

Cf. A000217, A022915, A135860, A135861, A135862.

Diagonal of A084546.

Without loops we have A116508, covering A367863, unlabeled A006649.

Allowing edges of any positive size gives A136556, covering A054780.

The covering case is A368597.

The unlabeled version is A368598, covering A368599.

The connected case is A368951.

A000666 counts unlabeled loop-graphs, covering A322700.

A006125 (shifted left) counts loop-graphs, covering A322661.

A006129 counts covering simple graphs, connected A001187.

A058891 counts set-systems, unlabeled A000612.

Cf. A000085, A000272, A057500, A333331, A368596, A368730.

Sequence in context: A257476 A218673 A230478 * A006963 A243426 A365438

Adjacent sequences: A014065 A014066 A014067 * A014069 A014070 A014071

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved