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A014068 a(n) = binomial(n*(n+1)/2, n).

%I A014068 #56 Aug 06 2025 09:05:07
%S A014068 1,1,3,20,210,3003,54264,1184040,30260340,886163135,29248649430,
%T A014068 1074082795968,43430966148115,1917283000904460,91748617512913200,
%U A014068 4730523156632595024,261429178502421685800,15415916972482007401455,966121413245991846673830,64123483527473864490450300
%N A014068 a(n) = binomial(n*(n+1)/2, n).
%C A014068 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
%C A014068 Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - _Gus Wiseman_, Jan 25 2024
%H A014068 Harvey P. Dale, <a href="/A014068/b014068.txt">Table of n, a(n) for n = 0..370</a>
%H A014068 R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H A014068 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphLoop.html">Graph Loop</a>.
%F A014068 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
%F A014068 For n > 0, a(n) = A022915(n)/A022915(n-1). - _Gerald McGarvey_, Jul 26 2004
%F A014068 a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - _Amarnath Murthy_, Jul 14 2005
%F A014068 a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - _Zerinvary Lajos_, Nov 30 2009
%F A014068 From _Peter Bala_, Feb 27 2020: (Start)
%F A014068 a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
%F A014068 Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
%F A014068 a(n) = A084546(n,n). - _Gus Wiseman_, Jan 25 2024
%F A014068 a(n) = [x^n] (1+x)^(n*(n+1)/2). - _Vaclav Kotesovec_, Aug 06 2025
%e A014068 From _Gus Wiseman_, Jan 25 2024: (Start)
%e A014068 The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
%e A014068   {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
%e A014068              {{1},{1,2}}  {{1},{2},{1,2}}
%e A014068              {{2},{1,2}}  {{1},{2},{1,3}}
%e A014068                           {{1},{2},{2,3}}
%e A014068                           {{1},{3},{1,2}}
%e A014068                           {{1},{3},{1,3}}
%e A014068                           {{1},{3},{2,3}}
%e A014068                           {{2},{3},{1,2}}
%e A014068                           {{2},{3},{1,3}}
%e A014068                           {{2},{3},{2,3}}
%e A014068                           {{1},{1,2},{1,3}}
%e A014068                           {{1},{1,2},{2,3}}
%e A014068                           {{1},{1,3},{2,3}}
%e A014068                           {{2},{1,2},{1,3}}
%e A014068                           {{2},{1,2},{2,3}}
%e A014068                           {{2},{1,3},{2,3}}
%e A014068                           {{3},{1,2},{1,3}}
%e A014068                           {{3},{1,2},{2,3}}
%e A014068                           {{3},{1,3},{2,3}}
%e A014068                           {{1,2},{1,3},{2,3}}
%e A014068 (End)
%t A014068 Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* _Harvey P. Dale_, May 27 2014 *)
%o A014068 (Sage) [(binomial(binomial(n+1, n-1), n)) for n in range(20)] # _Zerinvary Lajos_, Nov 30 2009
%o A014068 (Magma) [Binomial(Binomial(n+1,2), n): n in [0..40]]; // _G. C. Greubel_, Feb 19 2022
%o A014068 (Python)
%o A014068 from math import comb
%o A014068 def A014068(n): return comb(comb(n+1,2),n) # _Chai Wah Wu_, Jul 14 2024
%Y A014068 Cf. A000217, A022915, A135860, A135861, A135862.
%Y A014068 Diagonal of A084546.
%Y A014068 Without loops we have A116508, covering A367863, unlabeled A006649.
%Y A014068 Allowing edges of any positive size gives A136556, covering A054780.
%Y A014068 The covering case is A368597.
%Y A014068 The unlabeled version is A368598, covering A368599.
%Y A014068 The connected case is A368951.
%Y A014068 A000666 counts unlabeled loop-graphs, covering A322700.
%Y A014068 A006125 (shifted left) counts loop-graphs, covering A322661.
%Y A014068 A006129 counts covering simple graphs, connected A001187.
%Y A014068 A058891 counts set-systems, unlabeled A000612.
%Y A014068 Cf. A000085, A000272, A057500, A333331, A368596, A368730.
%K A014068 nonn
%O A014068 0,3
%A A014068 _N. J. A. Sloane_