cp's OEIS Frontend

A002050 Number of simplices in barycentric subdivision of n-simplex.

%I A002050 M3939 N1622 #93 Jul 02 2025 16:01:54
%S A002050 0,1,5,25,149,1081,9365,94585,1091669,14174521,204495125,3245265145,
%T A002050 56183135189,1053716696761,21282685940885,460566381955705,
%U A002050 10631309363962709,260741534058271801,6771069326513690645
%N A002050 Number of simplices in barycentric subdivision of n-simplex.
%C A002050 Stirling transform of A052849(n)=[1,4,12,48,240,...] is a(n)=[1,5,25,149,1081,...]. - _Michael Somos_, Mar 04 2004
%C A002050 Stirling transform of A000142(n-1)=[0,1,2,6,24,...] is a(n-1)=[0,1,5,25,149,...]. - _Michael Somos_, Mar 04 2004
%C A002050 Stirling transform of 2*A005359(n-1)=[1,0,4,0,48,0,...] is a(n-1)=[1,1,5,25,149,...]. - _Michael Somos_, Mar 04 2004
%C A002050 "Stirling-Bernoulli transform" of A000225. - _Paul Barry_, Apr 20 2005
%C A002050 a(n) is the number of nonempty words that can be formed from an alphabet of nonempty subsets of [n] so that the letters in each word are pairwise disjoint. - _Geoffrey Critzer_, Apr 12 2009
%C A002050 Row sums of A053440. - _Peter Bala_, Jul 12 2014
%C A002050 Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 5, 7, 5, 1, 5, 4, 5, 7, 5, 1, 5, 4, 5, 7, 5, 1, 5, 4, 5, 7, 5, ...], with an apparent period of 6 = phi(9) beginning at a(5). - _Peter Bala_, Aug 03 2023
%D A002050 R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
%D A002050 Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
%D A002050 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002050 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002050 T. D. Noe, <a href="/A002050/b002050.txt">Table of n, a(n) for n = 0..100</a>
%H A002050 R. Austin, R. K. Guy, & R. Nowakowski, <a href="/A000629/a000629.pdf">Unpublished notes, 1987</a>
%H A002050 Bérénice Delcroix-Oger and Clément Dupont, <a href="https://arxiv.org/abs/2505.06094">Lie-operads and operadic modules from poset cohomology</a>, arXiv:2505.06094 [math.CO], 2025. See p. 28.
%H A002050 R. K. Guy, <a href="/A002050/a002050_1.pdf">Letter to N. J. A. Sloane, Nov 21 1974</a>
%H A002050 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=149">Encyclopedia of Combinatorial Structures 149</a>
%H A002050 D. S. Kluk & N. J. A. Sloane, <a href="/A002050/a002050_3.pdf">Correspondence, 1979</a>
%H A002050 G. J. Simmons, <a href="http://www.jstor.org/stable/2689153">A combinatorial problem associated with a family of combination locks</a>, Math. Mag., 37 (1964), 127-132 (but there are errors).
%H A002050 G. J. Simmons, <a href="/A002050/a002050.pdf">A combinatorial problem associated with a family of combination locks</a>, Math. Mag., 37 (1964), 127-132 [Annotated, corrected, scanned copy]
%H A002050 G. J. Simmons, <a href="/A002050/a002050_2.pdf">Letter to N. J. Sloane, May 29 1974</a>
%H A002050 J. F. Steffensen, <a href="http://dx.doi.org/10.1080/03461238.1928.10416863">On a class of polynomials and their application to actuarial problems</a>, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.
%F A002050 E.g.f.: (exp(2x)-exp(x))/(2-exp(x)).
%F A002050 a(n) = A000629(n) - 1.
%F A002050 a(n) = Sum_{k=0..n} (-1)^(n-k)k!*S2(n, k)(2^k-1). - _Paul Barry_, Apr 20 2005
%F A002050 a(n) = Sum_{k=1...n} binomial(n,k)*A000670(k). - _Geoffrey Critzer_, Apr 12 2009
%F A002050 a(n) ~ n!/log(2)^(n+1). - _Vaclav Kotesovec_, Jul 29 2013
%F A002050 a(n) = 1 + 2*Sum_{k=2..n} k!*Stirling2(n,k), n > 0, a(0)=1. - _Vladimir Kruchinin_, Sep 27 2013
%F A002050 G.f.: T(0)/(1-2*x) - 1/(1-x), where T(k) = 1 - 2*x^2*(k+1)^2/(2*x^2*(k+1)^2 - (1 - 2*x - 3*x*k)*(1 - 5*x - 3*x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 29 2013
%F A002050 G.f.: Sum_{j>=1} j!*x^j / Product_{k=0..j} (1 - (k + 1)*x). - _Ilya Gutkovskiy_, Apr 04 2019
%t A002050 Table[Sum[Binomial[n, i]*Sum[StirlingS2[i, k]*k!, {k, 1, i}], {i, 1, n}], {n, 0, 20}] (* _Geoffrey Critzer_, Apr 12 2009 *)
%t A002050 With[{nn=20},CoefficientList[Series[(Exp[2x]-Exp[x])/(2-Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 28 2013 *)
%t A002050 a[0] = 0; a[n_] := 2*Sum[k!*StirlingS2[n, k], {k, 2, n}] + 1; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Sep 27 2013, after _Vladimir Kruchinin_ *)
%o A002050 (PARI) a(n)=if(n<0,0,n!*polcoeff(subst((y+y^2)/(1-y),y,exp(x+x*O(x^n))-1),n));
%Y A002050 A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - _N. J. A. Sloane_, Jul 04 2012
%Y A002050 A diagonal of the triangle in A241168. Row sums of A053440.
%K A002050 nonn,easy,nice
%O A002050 0,3
%A A002050 _N. J. A. Sloane_
%E A002050 More terms from _James Sellers_, Aug 22 2000