This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137775 #25 Apr 24 2025 09:44:40
%S A137775 1,0,3,6,45,252,1935,16146,153657,1616760,18699579,235498590,
%T A137775 3207570597,46968796404,735689606535,12272343940458,217191191400945,
%U A137775 4064131571557104,80166987477918963,1662468879466624950,36156426996107254941,822876672690142595820
%N A137775 Number of triples of permutations on n letters such that for each j, exactly one of the permutations fixes j and the other two have the same image on j.
%C A137775 This sequence arises in a calculation of the fourth moments of the volumes of random polytopes in certain very symmetric convex bodies.
%D A137775 M. Meckes, Volumens of symmetric random polytopes, Arch. Math. 82 (2004) 85--96.
%H A137775 Alois P. Heinz, <a href="/A137775/b137775.txt">Table of n, a(n) for n = 0..200</a>
%F A137775 a(n) = (n-1) * (a(n-1) + 3*a(n-2)) with a(0)=1. [corrected by _Seiichi Manyama_, Apr 23 2025]
%F A137775 E.g.f.: exp(-3x)/(1-x)^3.
%F A137775 a(n) is the number of derangements (permutations with no fixed points) of n elements where each cycle is colored with one of three colors. - _Michael Somos_, Jan 19 2011
%F A137775 G.f.: hypergeom([1,3],[],x/(1+3*x))/(1+3*x). - _Mark van Hoeij_, Nov 08 2011
%F A137775 a(n) ~ n! * exp(-3) * n^2/2. - _Vaclav Kotesovec_, Oct 08 2013
%F A137775 a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+2,2)/(n-k)!. - _Seiichi Manyama_, Apr 23 2025
%e A137775 a(2) = 3 because one of the permutations must be the identity and the other two are the transposition (1 2); there are three ways to pick which is the identity.
%e A137775 a(4) = 45 because there are 6 derangements with one 4-cycle with 3^1 ways to color each derangement and 3 derangements with two 2-cycles with 3^2 ways to color each derangement. - _Michael Somos_, Jan 19 2011
%t A137775 Range[0, 20]! CoefficientList[Series[Exp[ -3x]/(1 - x)^3, {x, 0, 20}], x]
%o A137775 (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -3 * x + x * O(x^n) ) / ( 1 - x )^3, n ) )} /* _Michael Somos_, Jan 19 2011 */
%Y A137775 Cf. A000007, A000166, A024000, A087981.
%K A137775 nonn
%O A137775 0,3
%A A137775 Mark W. Meckes (mark.meckes(AT)case.edu), May 06 2008
%E A137775 Added a(0)=1 by _Michael Somos_, Jan 19 2011