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A001005 Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.
(Formerly M1353 N0520)
1
1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) is also the number of rooted trees on n nodes such that each node has 0, 2, or 3 children. - Patrick Devlin, Mar 04 2012

REFERENCES

P Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 396

T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.

L. Smiley, a(7) and a(8)

Index entries for sequences related to rooted trees

FORMULA

G.f. for a(n+1) satisfies A(x)=x*(1+A(x)^2+A(x)^3). - Christian G. Bower, Dec 15 1999

a(n) = sum(((n)!/(k!*j!*(n-k-j+1)!)*[2*k+3*j=n], k=0..floor(n/2), j=0..floor(n/3)). - Len Smiley, Jun 18 2005

Recurrence: 2*(n+1)*(2*n+3)*(26*n+1)*a(n) = -(n-1)*(26*n^2 + 53*n + 18)*a(n-1) + 6*(n-1)*(78*n^2 + 42*n - 25)*a(n-2) + 31*(n-2)*(n-1)*(26*n+27)*a(n-3). - Vaclav Kotesovec, Aug 14 2013

a(n) ~ c*d^n/n^(3/2), where d = ((6371-624*sqrt(78))^(1/3)+(6371+624*sqrt(78))^(1/3)-1)/12 = 2.610718613276039349818649... is the root of the equation 4d^3 + d^2 - 18d - 31 = 0 and c = d^2 / (2*sqrt(Pi)*sqrt(1 + 3*d + sqrt(1 + 3*d))) = 0.559628309722556021604897336422272... - Vaclav Kotesovec, Aug 14 2013, updated Jun 27 2018

a(n) = sum_{k=1..floor(n/2)} C(n,k-1)*C(k,n-2k)/k, n>0. - Michael D. Weiner, Mar 02 2015

EXAMPLE

a(7)=21: 7 rotations of [12][34][567], 7 rotations of [12][45][367], 7 rotations of [12][37][456].

MAPLE

a:=proc(n::nonnegint) local k, j; a(n):=0; for k from 0 to floor(n/2) do for j from 0 to floor(n/3) do if (2*k+3*j=n) then a(n):=a(n)+(n)!/(k!*j!*(n-k-j+1)!) fi od od; print(a(n)) end proc; seq(a(i), i=0..30); # Len Smiley

MATHEMATICA

Table[Sum[(n)!/(k!*j!*(n - k - j + 1)!) * KroneckerDelta[2*k + 3*j - n], {k, 0, Floor[n/2]}, {j, 0, Floor[n/3]}], {n, 0, 20}] (* Ricardo Bittencourt, Jun 09 2013 *)

CoefficientList[ InverseSeries[x/(1+x^2+x^3) + O[x]^66]/x, x] (* Jean-Fran├žois Alcover, Feb 15 2016, after Joerg Arndt*)

PROG

(PARI) Vec(serreverse(x/(1+x^2+x^3)+O(x^66))/x) /* Joerg Arndt, Aug 19 2012 */

CROSSREFS

Sequence in context: A117647 A121568 A276464 * A009735 A177245 A283511

Adjacent sequences:  A001002 A001003 A001004 * A001006 A001007 A001008

KEYWORD

nonn,eigen

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms, formula and comment from Christian G. Bower, Dec 15 1999

Additional comments from Len Smiley, Jun 18 2005

STATUS

approved

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Last modified October 15 13:01 EDT 2018. Contains 316236 sequences. (Running on oeis4.)