A001454 -id:A001454 - cp's OEIS frontend

A001453 Catalan numbers - 1.

1, 4, 13, 41, 131, 428, 1429, 4861, 16795, 58785, 208011, 742899, 2674439, 9694844, 35357669, 129644789, 477638699, 1767263189, 6564120419, 24466267019, 91482563639, 343059613649, 1289904147323, 4861946401451, 18367353072151, 69533550916003, 263747951750359

Offset: 2

N. J. A. Sloane

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).

Alois P. Heinz, Table of n, a(n) for n = 2..500 (first 170 terms from Vincenzo Librandi)
R. M. Baer and P. Brock, Natural sorting over permutation spaces, Math. Comp. 22 1968 385-410.
J. M. Hammersley, A few seedings of research, in Proc. Sixth Berkeley Sympos. Math. Stat. and Prob., ed. L. M. le Cam et al., Univ. Calif. Press, 1972, Vol. I, pp. 345-394.
Piera Manara and Claudio Perelli Cippo, The fine structure of 4321 avoiding involutions and 321 avoiding involutions, PU. M. A. Vol. 22 (2011), 227-238.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Cf. A000108, A001454. Column k=2 of A047874.
A141364 is essentially the same sequence.
All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.

[Catalan(n)-1: n in [2..30]]; // Vincenzo Librandi, May 22 2011

with(combstruct): bin := {B=Union(Z,Prod(B,B))}: seq(count([B,bin, unlabeled], size=n+1)-1, n=2..30); # Zerinvary Lajos, Dec 05 2007

Array[CatalanNumber, 30, 2] - 1 (* Jean-François Alcover, Mar 11 2014 *)

combinat::dyckWords::count(n)-1 $ n = 2..26; // Zerinvary Lajos, May 08 2008

a(n)=(2*n)!/n!/(n+1)!-1 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = A000108(n) - 1 = binomial(2*n,n)/(n+1) - 1.
D-finite with recurrence: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(9*n-13)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Sep 04 2013
a(n) = Sum_{k=1..floor(n/2)} (C(n,k)-C(n,k-1))^2. - J. M. Bergot, Sep 17 2013
a(n) = Sum_{k=1..n-1} A000245(n-k-1). - John M. Campbell, Dec 28 2016
From Ilya Gutkovskiy, Dec 28 2016: (Start)
O.g.f.: (1 - sqrt(1 - 4*x))/(2*x) - 1/(1 - x).
E.g.f.: exp(x)*(exp(x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1). (End)
a(n)= 3*Sum_{k=1..n} binomial(2*k-2,k)/(k+1). - Gary Detlefs, Feb 14 2020

More terms from James Sellers, Sep 08 2000

A047874 Triangle of numbers T(n,k) = number of permutations of (1,2,...,n) with longest increasing subsequence of length k (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 9, 1, 1, 41, 61, 16, 1, 1, 131, 381, 181, 25, 1, 1, 428, 2332, 1821, 421, 36, 1, 1, 1429, 14337, 17557, 6105, 841, 49, 1, 1, 4861, 89497, 167449, 83029, 16465, 1513, 64, 1, 1, 16795, 569794, 1604098, 1100902, 296326, 38281, 2521, 81, 1

Offset: 1

Eric Rains (rains(AT)caltech.edu)

Mirror image of triangle in A126065.
T(n,m) is also the sum of squares of n!/(product of hook lengths), summed over the partitions of n in exactly m parts (Robinson-Schensted correspondence). - Wouter Meeussen, Sep 16 2010
Table I "Distribution of L_n" on p. 98 of the Pilpel reference. - Joerg Arndt, Apr 13 2013
In general, for column k is a(n) ~ product(j!, j=0..k-1) * k^(2*n+k^2/2) / (2^((k-1)*(k+2)/2) * Pi^((k-1)/2) * n^((k^2-1)/2)) (result due to Regev) . - Vaclav Kotesovec, Mar 18 2014

			T(3,2) = 4 because 132, 213, 231, 312 have longest increasing subsequences of length 2.
Triangle T(n,k) begins:
  1;
  1,   1;
  1,   4,    1;
  1,  13,    9,    1;
  1,  41,   61,   16,   1;
  1, 131,  381,  181,  25,  1;
  1, 428, 2332, 1821, 421, 36,  1;
  ...

Leonid Petrov, Rows n = 1..120, flattened (first 60 rows from Alois P. Heinz)
P. Diaconis, Group Representations in Probability and Statistics, IMS, 1988; see p. 112.
FindStat - Combinatorial Statistic Finder, The length of the longest increasing subsequence of the permutation.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
J. M. Hammersley, A few seedings of research, in Proc. Sixth Berkeley Sympos. Math. Stat. and Prob., ed. L. M. le Cam et al., Univ. Calif. Press, 1972, Vol. I, pp. 345-394.
Guo-Niu Han, A promenade in the garden of hook length formulas, Slides, 61st SLC Curia, Portugal - September 22, 2008. [From _Wouter Meeussen_, Sep 16 2010]
J. Hunt and T. Szymanski, A fast algorithm for computing longest common subsequences, Commun. ACM, 20 (1977), 350-353.
E. Irurozki, B. Calvo, and J. A. Lozano, Sampling and learning the Mallows model under the Ulam distance, Technical Report, 2014.
S. Pilpel, Descending subsequences of random permutations, J. Combin. Theory, A 53 (1990), 96-116.
A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), 115-136.
C. Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13 (1961), 179-191.
Richard P. Stanley, Increasing and Decreasing Subsequences of Permutations and Their Variants, arXiv:math/0512035 [math.CO], 2005.
Wikipedia, Longest increasing subsequence problem

Cf. A047887 and A047888.
Columns k=1-10 give: A000012, A001453, A001454, A001455, A001456, A001457, A001458, A239432, A245665, A245666.
Row sums give A000142.
Cf. A047884. - Wouter Meeussen, Sep 16 2010
Cf. A224652 (Table II "Distribution of F_n" on p. 99 of the Pilpel reference).
Cf. A245667.
T(2n,n) gives A267433.
Cf. A003316.

h:= proc(l) local n; n:= nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= n-> seq(g(n-k, min(n-k, k), [k]), k=1..n):
seq(T(n), n=1..12);  # Alois P. Heinz, Jul 05 2012

Table[Total[NumberOfTableaux[#]^2&/@ IntegerPartitions[n,{k}]],{n,7},{k,n}] (* Wouter Meeussen, Sep 16 2010, revised Nov 19 2013 *)
h[l_List] := Module[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := Table[g[n-k, Min[n-k, k], {k}], {k, 1, n}]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A003316(n). - Alois P. Heinz, Nov 04 2018

A060213 Lesser of twin primes whose average is 6 times a prime.

Original entry on oeis.org

11, 17, 29, 41, 101, 137, 281, 617, 641, 821, 1697, 1877, 2081, 2237, 2381, 2657, 2801, 3461, 3557, 3917, 4637, 4721, 5441, 6197, 6701, 8537, 8597, 9677, 10937, 12161, 12377, 12821, 12917, 13217, 13721, 13757, 13997, 14081, 16061, 17417

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Lowest factor-density among all positive consecutive integer triples; for p > 41, last digit of p can be only 1 or 7 (see Alexandrov link, p. 15). - Lubomir Alexandrov, Nov 25 2001

			102197 is here because 102198 = 17033*6 and 17033 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998, see p. 15.

Cf. A002822, A058383, A059960, A027856, A001454, A001359, A060212.

map(t -> 6*t-1, select(p -> isprime(p) and isprime(6*p-1) and isprime(6*p+1), [2,seq(i,i=3..10000,2)]));

Transpose[Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]] == 2 && PrimeQ[Mean[#]/6]&]][[1]] (* Harvey P. Dale, May 04 2014 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 6) && isprime((n+1)/6); \\ Michel Marcus, Dec 14 2013

a(n) = 6 * A060212(n) - 1. - Sean A. Irvine, Oct 31 2022

Offset changed to 1 by Michel Marcus, Dec 14 2013

A060210 Largest prime factor of 1+smaller term of twin primes.

Original entry on oeis.org

2, 3, 3, 3, 5, 7, 5, 3, 17, 3, 23, 5, 5, 3, 11, 19, 5, 5, 47, 13, 29, 7, 3, 11, 29, 19, 5, 103, 107, 11, 5, 137, 23, 13, 7, 17, 43, 7, 59, 13, 3, 41, 71, 43, 31, 11, 17, 11, 19, 31, 67, 5, 139, 283, 41, 149, 13, 313, 23, 13, 37, 13, 347, 29, 11, 71, 17, 373, 7, 11, 13, 397, 17

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Also: Largest prime factor of the average, or the sum, of twin prime pairs. - M. F. Hasler, Jan 03 2011

			101 is the 9th lesser twin, 102 = 2*3*17, and its max p factor is 17=a(9).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A001359, A001454, A002822, A059960, A027856.

FactorInteger[1+#][[-1,1]]&/@Select[Partition[Prime[Range[500]],2,1], #[[2]]- #[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 16 2017 *)

p=3; for(n=1,1e3, until(o+2==p,p=nextprime(2+o=p)); print1(vecmax(factor(p-1)[,1])","))  \\ M. F. Hasler, Jan 03 2011

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

Original entry on oeis.org

7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633, 578415690713089, 1141260857376769, 57711166318706689

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Larger of twin primes p such that p-1 = (2^u)*(3^w), u,w >= 1.

			a(4) = 73, {71,73} are twin primes and (71 + 73)/2 = 72 = 2*2*2*3*3.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000)
Harsh Aggarwal, Table of n, a(n) for n = 62..91 (terms from 10^1000 to 10^20000)

Cf. A001454, A002822, A027856, A033845, A058383, A059960.

Take[Select[Sort[Flatten[Table[2^a 3^b,{a,250},{b,250}]]],AllTrue[#+{1,-1},PrimeQ]&]+1,23] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2019 *)

isok(p) = isprime(p) && isprime(p-2) && (vecmax(factor(p-1)[,1]) == 3); \\ Michel Marcus, Sep 05 2017

a(n) = A027856(n+1) + 1. - Amiram Eldar, Mar 17 2025

Name corrected by Sean A. Irvine, Oct 31 2022

A239432 Number of permutations of length n with longest increasing subsequence of length 8.

Original entry on oeis.org

1, 64, 2521, 79861, 2250887, 59367101, 1508071384, 37558353900, 927716186325, 22904111472825, 568209449266202, 14216730315766814, 359666061054003144, 9216708503647774264, 239524408949706575548, 6317740398995612513164, 169207499997274346326579, 4602911809939402715164066

Offset: 8

Vaclav Kotesovec, Mar 18 2014

nonn

In general, for column k of A047874 is a(n) ~ product(j!, j=0..k-1) * k^(2*n+k^2/2) / (2^((k-1)*(k+2)/2) * Pi^((k-1)/2) * n^((k^2-1)/2)) [Regev, 1981].

Vaclav Kotesovec, Table of n, a(n) for n = 8..135
A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), 115-136.

Cf. A047874, A001453, A001454, A001455, A001456, A001457, A001458.

a(n) ~ 1913625 * 2^(6*n+77) / (Pi^(7/2) * n^(63/2)).

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A001453 Catalan numbers - 1.

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A047874 Triangle of numbers T(n,k) = number of permutations of (1,2,...,n) with longest increasing subsequence of length k (1<=k<=n).

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A060213 Lesser of twin primes whose average is 6 times a prime.

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A060210 Largest prime factor of 1+smaller term of twin primes.

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A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

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A239432 Number of permutations of length n with longest increasing subsequence of length 8.

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