

A003517


Number of permutations of [n+1] with exactly 1 increasing subsequence of length 3.
(Formerly M4177)


42



1, 6, 27, 110, 429, 1638, 6188, 23256, 87210, 326876, 1225785, 4601610, 17298645, 65132550, 245642760, 927983760, 3511574910, 13309856820, 50528160150, 192113383644, 731508653106, 2789279908316, 10649977831752, 40715807302800
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OFFSET

2,2


COMMENTS

a(n4) = number of nth generation vertices in the tree of sequences with unit increase labeled by 5 (cf. Zoran Sunic reference).  Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+3,n2).  Emeric Deutsch, May 30 2004
a(n) = A214292(2*n,n3) for n > 2.  Reinhard Zumkeller, Jul 12 2012
a(n) is the number of NorthEast paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly once. By symmetry, it is also the number of NorthEast paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly once. Details can be found in Section 3.3 in Pan and Remmel's link.  Ran Pan, Feb 02 2016
a(n) is the number of permutations pi of [n+3] such that s(pi)=321456...(n+3), where s denotes West's stacksorting map.  Colin Defant, Jan 14 2019
a(n) is also the number of permutations of [n+1] avoiding the pattern 321. For permutations avoiding any of the other permutations of [3] (that is, any of 132, 213, 231, or 312) see A002054.  N. J. A. Sloane, Nov 26 2022


REFERENCES

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 = 2..1000
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multirooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
David Callan, A recursive bijective approach to counting permutations..., arXiv:math/0211380 [math.CO], 2002.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743751. [Annotated scanned copy]
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21 , No. 2 (2010), pp. 265284 (see 4.3 p. 277).
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395405.
Markus Kuba and Alois Panholzer, Stirling permutations containing a single pattern of length three, Australasian Journal of Combinatorics, Vol. 74, No. 2 (2019), pp. 216239.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq., Vol. 13 (2010), Article 10.7.4.
John Noonan and Doron Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
John Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math., Vol. 152, No. 13 (1996), pp. 307313.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 8390.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 8390. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., Vol. 10 (2003), Article N5.


FORMULA

a(n) = 6*binomial(2*n+1, n2)/(n+4).
G.f.: x^2*C(x)^6, where C(x) is g.f. for the Catalan numbers (A000108).  Emeric Deutsch, May 30 2004
E.g.f.: exp(2*x)*(Bessel_I(2,2*x)  Bessel_I(4,2*x)).  Paul Barry, Jun 04 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i1]=1, A[i,j]=Catalan(ji), (i<=j), and A[i,j]=0, otherwise. Then, for n >= 5, a(n3) = (1)^(n5)*coeff(charpoly(A,x),x^5).  Milan Janjic, Jul 08 2010
a(n) = Sum_{i>=1, j>=1, k>=1, i+j+k=n+1} Catalan(i)*Catalan(j)*Catalan(k). T. D. Noe, Dec 22 2010
Dfinite with recurrence (n+4)*(n2)*a(n) + 2*n*(2*n+1)*a(n1) = 0.  R. J. Mathar, Dec 04 2012
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=2} 1/a(n) = 7/2  34*Pi/(27*sqrt(3)).
Sum_{n>=2} (1)^n/a(n) = 828*log(phi)/(25*sqrt(5))  2819/450, where phi is the golden ratio (A001622). (End)


EXAMPLE

a(3)=6 because the only permutations of 1234 with exactly 1 increasing subsequence of length 3 are 1423, 4123, 1342, 2314, 2341, 3124.


MAPLE

A003517List := proc(m) local A, P, n; A := [1]; P := [1, 1, 1, 1, 1];
for n from 1 to m  2 do P := ListTools:PartialSums([op(P), P[1]]);
A := [op(A), P[1]] od; A end: A003517List(25); # Peter Luschny, Mar 26 2022


MATHEMATICA

f[x_] = (Sqrt[1  4 x]  1)^6/(64 x^4); CoefficientList[Series[f[x], {x, 0, 25}], x][[3 ;; 26]] (* JeanFrançois Alcover, Jul 13 2011, after g.f. *)
Table[6 Binomial[2n+1, n2]/(n+4), {n, 2, 30}] (* Harvey P. Dale, Feb 27 2012 *)


PROG

(PARI) a(n)=6*binomial(2*n+1, n2)/(n+4) \\ Charles R Greathouse IV, May 18 2015
(PARI) x='x+O('x^50); Vec(x^2*((1(14*x)^(1/2))/(2*x))^6) \\ Altug Alkan, Nov 01 2015


CROSSREFS

T(n, n+6) for n=0, 1, 2, ..., array T as in A047072.
See also A002054.
Cf. A001089, A084249, A000108, A000245, A002057, A000344, A000588, A003518, A003519, A001392, A001622.
First differences are in A026017.
A diagonal of any of the essentially equivalent arrays: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Sequence in context: A094788 A221863 A216263 * A108958 A005284 A198694
Adjacent sequences: A003514 A003515 A003516 * A003518 A003519 A003520


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



