A002050 Number of simplices in barycentric subdivision of n-simplex. (Formerly M3939 N1622)

0, 1, 5, 25, 149, 1081, 9365, 94585, 1091669, 14174521, 204495125, 3245265145, 56183135189, 1053716696761, 21282685940885, 460566381955705, 10631309363962709, 260741534058271801, 6771069326513690645

Offset

0,3

Comments

Stirling transform of A052849(n)=[1,4,12,48,240,...] is a(n)=[1,5,25,149,1081,...]. - Michael Somos, Mar 04 2004

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

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

"Stirling-Bernoulli transform" of A000225. - Paul Barry, Apr 20 2005

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

Row sums of A053440. - Peter Bala, Jul 12 2014

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

References

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..100

R. Austin, R. K. Guy, & R. Nowakowski, Unpublished notes, 1987

R. K. Guy, Letter to N. J. A. Sloane, Nov 21 1974

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 149

D. S. Kluk & N. J. A. Sloane, Correspondence, 1979

G. J. Simmons, A combinatorial problem associated with a family of combination locks, Math. Mag., 37 (1964), 127-132 (but there are errors).

G. J. Simmons, A combinatorial problem associated with a family of combination locks, Math. Mag., 37 (1964), 127-132 [Annotated, corrected, scanned copy]

G. J. Simmons, Letter to N. J. Sloane, May 29 1974

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.

Formula

E.g.f.: (exp(2x)-exp(x))/(2-exp(x)).

a(n) = A000629(n) - 1.

a(n) = Sum_{k=0..n} (-1)^(n-k)k!*S2(n, k)(2^k-1). - Paul Barry, Apr 20 2005

a(n) = Sum_{k=1...n} binomial(n,k)*A000670(k). - Geoffrey Critzer, Apr 12 2009

a(n) ~ n!/log(2)^(n+1). - Vaclav Kotesovec, Jul 29 2013

a(n) = 1 + 2*Sum_{k=2..n} k!*Stirling2(n,k), n > 0, a(0)=1. - Vladimir Kruchinin, Sep 27 2013

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

G.f.: Sum_{j>=1} j!*x^j / Product_{k=0..j} (1 - (k + 1)*x). - Ilya Gutkovskiy, Apr 04 2019

Mathematica

Table[Sum[Binomial[n, i]*Sum[StirlingS2[i, k]*k!, {k, 1, i}], {i, 1, n}], {n, 0, 20}] (* Geoffrey Critzer, Apr 12 2009 *)
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 *)
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 *)

Prog

(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst((y+y^2)/(1-y), y, exp(x+x*O(x^n))-1), n));

Crossrefs

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

A diagonal of the triangle in A241168. Row sums of A053440.

Sequence in context: A121639 A098349 A098212 * A047782 A106565 A200031

Adjacent sequences: A002047 A002048 A002049 * A002051 A002052 A002053

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Aug 22 2000

Status

approved