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A014068
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a(n) = binomial(n*(n+1)/2, n).
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38
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1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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Eric Weisstein's World of Mathematics, Graph Loop.
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FORMULA
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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
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
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)
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EXAMPLE
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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)
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MATHEMATICA
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Binomial[First[#], Last[#]]&/@With[{nn=20}, Thread[{Accumulate[ Range[ 0, nn]], Range[ 0, nn]}]] (* Harvey P. Dale, May 27 2014 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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