cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A001181 Number of Baxter permutations of length n (also called Baxter numbers).

Original entry on oeis.org

1, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586, 105791986682, 687782586844, 4517543071924, 29949238543316, 200234184620736, 1349097425104912, 9154276618636016, 62522506583844272
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

As shown by Dulucq and Guilbert (see for example "Baxter Permutations", Discrete Math., 1998), a(n) is also the number of possible paths for three vicious walkers of length n-1 (a.k.a. "vicious 3-watermelons") [Essam & Guttmann (1995), Eq. (63)] [Jensen (2017), Eqs. (1), (2)]. The proof follows easily by comparing Ollerton's recurrence here and the recurrence in Eq. (60) of Essam & Guttmann. In fact, as discussed by Dulucq and Guilbert, this interpretation of the sequence has been known for a long time. - N. J. A. Sloane, Mar 19 2021; additional references provided by Olivier Gerard, Mar 22 2021.
From Roger Ford, Apr 11 2020: (Start)
a(n) is also the number of meanders with n top arches that as the number of arches is reduced by combining the first and last arches every even number of arches produces a meander.
Example: For n = 4, this meander has this trait.
/\ arches= 8
/\ / \
/ \ --> /\ //\ \
Starting meander: /\ /\ / /\ \ Split and /\/\//\\ ///\\/\\
\ \/ \ \/ / / rotate
\ \ / / bottom arches /\
\ \/ / / \ arches= 7
\ / / /\\ /\
\ / Combine start //\//\\\ //\\/\
\ / first arch with
meander: \/ end of last arch
/\ /\
/\ / \ Combine arches /\ //\\ arches= 6
\ \ / /\ \ again then /\ //\\ ///\\\
\ \ \/ / / rotate and join
\ \ / / <-- /\
\ \/ / //\\ /\ arches= 5
\ / Combine ///\\\ //\\
\/ /\
meander: / \
/ /\ \ Combine /\
\/ \/ <-- //\\ /\/\ arches= 4
/\
Combine /\ //\\ arches= 3
meander: /\ Combine
\/ <-- /\ /\ arches= 2
(End)

Examples

			G.f. = x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 422*x^6 + 2074*x^7 + ...
a(4) = 22 since all permutations of length 4 are Baxter except 2413 and 3142. - _Michael Somos_, Jul 19 2002

References

  • Arthur T. Benjamin and Naiomi T. Cameron, Counting on determinants, American Mathematical Monthly, 112.6 (2005): 481-492.
  • W. M. Boyce, Generation of a class of permutations associated with commuting functions, Math. Algorithms, 2 (1967), 19-26.
  • William M. Boyce, Commuting functions with no common fixed point, Transactions of the American Mathematical Society 137 (1969): 77-92.
  • T. Y. Chow, Review of "Bonichon, Nicolas; Bousquet-Mélou, Mireille; Fusy, Éric; Baxter permutations and plane bipolar orientations. Sem. Lothar. Combin. 61A (2009/10), Art. B61Ah, 29 pp.", MathSciNet Review MR2734180 (2011m:05023).
  • S. Dulucq and O. Guibert, Baxter permutations, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995).
  • J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), 1995, pp. 5849ff. See (60) and (63).
  • Iwan Jensen, Three friendly walkers, Journal of Physics A: Mathematical and Theoretical, Volume 50:2 (2017), #24003, 14 pages; https://doi.org/10.1088/1751-8121/50/2/024003. See (1), (2).
  • S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399, Table A.7.
  • 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).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.55.
  • Doron Zeilberger, The method of creative telescoping, J. Symb. Comput. 11.3 (1991): 195-204. See Section 7.8.

Links

Crossrefs

Cf. A001183, A001185, A046996, A006002, A007318, A056939, A359363.

Programs

  • GAP
    Concatenation([1], List([1..30], n-> 2*Sum([1..n], k-> Binomial(n+1,k-1)* Binomial(n+1,k)*Binomial(n+1, k+1) )/(n*(n+1)^2) )); # G. C. Greubel, Jul 24 2019
  • Haskell
    a001181 0 = 1
a001181 n =
   (sum $ map (\k -> product $ map (a007318 (n+1)) [k-1..k+1]) [1..n])
    `div` (a006002 n)
-- Reinhard Zumkeller, Oct 23 2011
  • Magma
    [1] cat [2*(&+[Binomial(n+1,k-1)*Binomial(n+1,k)* Binomial(n+1, k+1): k in [1..n]])/(n*(n+1)^2): n in [1..30]]; // G. C. Greubel, Jul 24 2019
  • Maple
    C := binomial; A001181 := proc(n) local k; add(C(n+1, k-1)*C(n+1, k)* C(n+1, k+1)/ (C(n+1, 1)*C(n+1, 2)), k = 1..n); end;
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1,
      ((7*n^2+7*n-2)*a(n-1)+8*(n-1)*(n-2)*a(n-2))/((n+2)*(n+3)))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
    A001181[n_] := HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -1] (* Richard L. Ollerton, Sep 13 2006 *)
a[0]=1; a[1]=1; a[n_] := a[n] = ((7n^2+7n-2)*a[n-1] + 8(n-1)(n-2)*a[n-2]) / ((n+2)(n+3)); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 28 2015, from 3rd formula *)
  • PARI
    {a(n) = if( n<0, 0, sum( k=1, n, binomial(n+1, k-1) * binomial(n+1, k) * binomial(n+1, k+1) / (binomial(n+1, 1) * binomial(n+1, 2))))}; /* Michael Somos, Jul 19 2002 */
  • Python
    from sympy import binomial as C
def a(n): return sum([(C(n + 1, k - 1)*C(n + 1, k)*C(n + 1, k + 1))/(C(n + 1, 1) * C(n + 1, 2)) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 25 2017
  • Sage
    [1]+[2*sum(binomial(n+1,k-1)*binomial(n+1,k)*binomial(n+1, k+1) for k in (1..n))/(n*(n+1)^2) for n in (1..30)] # G. C. Greubel, Jul 24 2019
print([BaxterPermutations(n).cardinality() for n in range(25)])
# Peter Luschny, May 21 2024

Formula

a(n) = Sum_{k=1..n} C(n+1,k-1) * C(n+1,k) * C(n+1,k+1) / (C(n+1,1) * C(n+1,2)).
(n + 1)*(n + 2)*(n + 3)*(3*n - 2)*a(n) = 2*(n + 1)*(9*n^3 + 3*n^2 - 4*n + 4)*a(n - 1) + (3*n - 1)*(n - 2)*(15*n^2 - 5*n - 14)*a(n - 2) + 8*(3*n + 1)*(n - 2)^2*(n - 3)*a(n - 3), if n>1. [Stanley, 1999] - Michael Somos, Jul 19 2002
From Richard L. Ollerton, Sep 13 2006: (Start)
D-finite with recurrence (n + 2)*(n + 3)*a(n) = (7*n^2 + 7*n - 2)*a(n-1) + 8*(n - 1)*(n - 2)*a(n-2), a(0) = a(1) = 1.
a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -1). (End)
[The coefficients of the polynomials p(n, x) = hypergeom([-1-n, -n, 1-n], [2, 3], -x) are given by the triangular form of A056939. - Peter Luschny, Dec 28 2022]
G.f.: -1 + (1/(3*x^2)) * (x-1 + (1-2*x)*hypergeom([-2/3, 2/3],[1],27*x^2/(1-2*x)^3) - (8*x^3-11*x^2-x)*hypergeom([1/3, 2/3],[2],27*x^2/(1-2*x)^3)/(1-2*x)^2 ). - Mark van Hoeij, Oct 23 2011
a(n) ~ 2^(3*n+5)/(Pi*sqrt(3)*n^4). - Vaclav Kotesovec, Oct 01 2012
0 = +a(n)*(+a(n+1)*(+512*a(n+2) + 2624*a(n+3) - 960*a(n+4)) +a(n+2)*(-1216*a(n+2) + 3368*a(n+3) - 1560*a(n+4)) + a(n+3)*(+600*a(n+3) + 120*a(n+4))) + a(n+1)*(+a(n+1)*(-64*a(n+2) - 1288*a(n+3) + 600*a(n+4)) +a(n+2)*(-136*a(n+2) + 295*a(n+3) - 421*a(n+4)) +a(n+3)*(+161*a(n+3) + 41*a(n+4))) + a(n+2)*(+a(n+2)*(+72*a(n+2) + 17*a(n+3) - 19*a(n+4)) + a(n+3)*(-a(n+3) - a(n+4))) if n>=0. - Michael Somos, Mar 09 2017
G.f.: ((x^3+3*x^2+3*x+1)/(1-8*x)^(3/4)*hypergeom([1/4, 5/4],[2],64*x*(1+x)^3/(8*x-1)^3) - 1 + x)/(3*x^2). - Mark van Hoeij, Nov 05 2023

Extensions

Changed initial term to a(0) = 1 (it was a(0) = 0, but there were compelling reasons to change it). - N. J. A. Sloane, Sep 14 2021

A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 50, 20, 1, 1, 35, 175, 175, 35, 1, 1, 56, 490, 980, 490, 56, 1, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1, 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1
Offset: 0

Views

Author

Mitch Harris

Keywords

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20This%20array%20is%20the%20main%20subject%20of%20the%20long%20article%20by%20Felsner%20et%20al.%20(2011).%20-%20_N.%20J.%20A.%20Sloane">3; cf. A342889. This array is the main subject of the long article by Felsner et al. (2011). - _N. J. A. Sloane, Apr 03 2021
This triangle is mentioned by Hoggatt (1977). - N. J. A. Sloane, Mar 27 2021
Determinants of 3 X 3 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005
Also determinants of 3 X 3 arrays whose entries come from a single row: T(n,k) = det [C(n,k),C(n,k-1),C(n,k-2); C(n,k+1),C(n,k),C(n,k-1); C(n,k+2),C(n,k+1),C(n,k)]. - Peter Bala, May 10 2012
From Gary W. Adamson, Jul 10 2012: (Start)
The triangular view of this triangle is
1;
1, 1;
1, 4, 1;
1, 10, 10, 1;
1, 20, 50, 20, 1;
The n-th row of this triangle is generated by applying the ConvOffs transform to the first n terms of 1, 4, 10, 20, ... (A000292 without leading zero). See A214281 for a procedural definition of the transformation and search "ConvOffs" for more examples. (End)
Define polynomials p(n, x) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -x). If the triangle is extended by the diagonal 1, 0, 0,... on the right side the resulting (0, 0)-based triangle is T*(n, k) = [x^k] p(n, x). The polynomials evaluated at x = 1 gives the number of Baxter permutations of length n (see the formula given by Richard L. Ollerton in A001181). - Peter Luschny, Dec 28 2022

Examples

			The initial rows of the array are:
     1      1      1      1      1      1 ...
     1      4     10     20     35     56 ...
     1     10     50    175    490   1176 ...
     1     20    175    980   4116  14112 ...
     1     35    490   4116  24696 116424 ...
     1     56   1176  14112 116424 731808 ...
     ...
Considered as a triangle, the initial rows are:
  [1],
  [1, 1],
  [1, 4, 1],
  [1, 10, 10, 1],
  [1, 20, 50, 20, 1],
  [1, 35, 175, 175, 35, 1],
  [1, 56, 490, 980, 490, 56, 1],
  [1, 84, 1176, 4116, 4116, 1176, 84, 1],
  [1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1],
  [1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1],
  [1, 220, 9075, 108900, 457380, 731808, 457380, 108900, 9075, 220, 1]
  ...

References

  • Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
  • R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1

Links

Crossrefs

Cf. A000372, A056932, A001263, A056940, A056941.
Antidiagonals sum to A001181 (Baxter permutations). Cf. A197208.
Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1..12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

Programs

  • Maple
    # To get initial terms of the array - N. J. A. Sloane, Apr 20 2021
bb := (k,l) -> binomial(k+l,k)*binomial(k+l+1,k)*binomial(k+l+2,k)*2/((k+1)^2*(k+2));
for k from 0 to 8 do
lprint([seq(bb(k,l),l=0..8)]);
od:
  • Mathematica
    t[n_, m_] = 2*Binomial[n, m]*Binomial[n + 1, m + 1]* Binomial[n + 2, m + 2]/((n - m + 1)^2*(n - m + 2)); Flatten[Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]] (* Roger L. Bagula, Jan 28 2009 *)
  • PARI
    \\ cf. A359363
C=binomial;
T(n,k)=if(n==0&&k==0,1,(C(n+1,k-1)*C(n+1,k)*C(n+1,k+1))/(C(n+1,1)*C(n+1,2)));
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 04 2024

Formula

Product_{k=0..2} binomial(n+m+k, m+k)/binomial(n+k, k) gives the array as a square.
T(n,m) = 2*binomial(n, m)*binomial(n+1, m+1)*binomial(n+2, m+2)/((n-m+1)^2*(n-m+2)). - Roger L. Bagula, Jan 28 2009
From Peter Bala, Oct 13 2011: (Start)
T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k)*C(n+2,k+2)*C(n+3,k+1);
T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k+1)*C(n+2,k)*C(n+3,k+2). Cf. A197208.
T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).
Define a(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is a(r,0)*a(r,n)/(a(r,k)*a(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)
The column generating functions of the square array (starting at column 1) are 1/(1 - x)^4, (1 + 3*x + x^2)/(1 - x)^7, (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1 - x)^10, ..., where the numerator polynomials are the row polynomials of A087647. See Barry p. 31. - Peter Bala, Oct 18 2023

A046996 Number of Baxter permutations: A001181/2.

Original entry on oeis.org

1, 3, 11, 46, 211, 1037, 5377, 29101, 163120, 941480, 5570280, 33664996, 207249719, 1296670793, 8229378293, 52895993341, 343891293422, 2258771535962, 14974619271658, 100117092310368, 674548712552456, 4577138309318008, 31261253291922136, 214800030086785976
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A359363.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, 2*n-3,
      ((7*n^2+7*n-2)*a(n-1)+8*(n-1)*(n-2)*a(n-2))/((n+2)*(n+3)))
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
    Sum[Binomial[n+1, k-1] Binomial[n+1, k] Binomial[n+1, k+1], {k, 1, n}]/(2 Binomial[n+1, 1] Binomial[n+1, 2])
  • SageMath
    def A046996(n): return PolyA359363(n, 1) // 2
print([A046996(n) for n in range(2, 26)])  # Peter Luschny, Jan 04 2024

Formula

a(n) = (1/2)*B(n, 1) where B(n, x) are the Baxter polynomials with coefficients A359363. - Peter Luschny, Jan 04 2024

Extensions

More terms from Olivier Gérard, Jun 26 2001

A368708 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -2).

Original entry on oeis.org

1, 1, 3, 13, 69, 417, 2763, 19609, 146793, 1146833, 9278595, 77292261, 659973933, 5756169681, 51137399979, 461691066417, 4228199347281, 39216540096993, 367890444302787, 3486697883136957, 33353178454762389, 321754445379041601, 3127955713554766923, 30624486778208481993, 301790556354721667769, 2991957347531210976817
Offset: 0

Views

Author

Joerg Arndt, Jan 04 2024

Keywords

Crossrefs

Cf. A001181, A007724, A056939, A217800, A359363, A368709.

Programs

  • Maple
    seq(simplify( hypergeom([-1 - n, -n, 1 - n], [2, 3], -2) ), n = 0..25); # Peter Bala, Sep 09 2024
  • Mathematica
    Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -2], {n, 0, 30}] (* Vaclav Kotesovec, Jan 04 2024 *)
a[0] := 1; a[n_] := 2^(n + 1)/(n*(n + 1)^2)*Sum[(1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k, 1, n}]; Table[a[n], {n, 0, 25}] (* Detlef Meya, May 28 2024 *)
  • Python
    def A368708(n):
    if n == 0: return 1
    return sum(2**k * v for k, v in enumerate(A359363Row(n))) // 2
print([A368708(n) for n in range(26)]) # Peter Luschny, Jan 04 2024
  • SageMath
    def A368708(n): return PolyA359363(n, 2) // 2 if n > 0 else 1
print([A368708(n) for n in range(23)])  # Peter Luschny, Jan 04 2024

Formula

a(n) = (1/2)*B(n, 2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - Peter Luschny, Jan 04 2024
a(n) ~ 3^(n + 7/6) * (2^(2/3) + 2^(1/3) + 1)^(n + 5/3) / (2^(4/3) * Pi * n^4). - Vaclav Kotesovec, Jan 04 2024
a(0) = 1, a(n) = 2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - Detlef Meya, May 29 2024
From Peter Bala, Sep 09 2024: (Start)
a(n+1) = Sum_{k = 0..n} A056939(n, k)*2^k.
P-recursive: (n+1)*(n+3)*(n+2)*(3*n-2)*a(n) = 3*(9*n^3+3*n^2-4*n+4)*(n+1)*a(n-1) + 3*(n-2)*(3*n-1)*(9*n^2-3*n-10)*a(n-2) + 27*(3*n+1)*(n-3)*(n-2)^2*a(n-3) = 0 with a(0) = 1, a(1) = 1 and a(2) = 3. (End)

A368709 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], +2).

Original entry on oeis.org

1, 1, -1, -3, 13, 17, -241, 121, 5081, -13327, -106705, 609589, 1850661, -23392159, -6796193, 811545073, -1688514383, -25224774367, 123764707231, 650087614573, -6385330335427, -9591188592399, 279171512779759, -318526766092183, -10665705513959287, 40625771132796817
Offset: 0

Views

Author

Joerg Arndt, Jan 04 2024

Keywords

Crossrefs

Cf. A368708, A001181, A007724, A217800, A359363.

Programs

  • Mathematica
    Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, 2], {n, 0, 30}] (* Vaclav Kotesovec, Jan 04 2024 *)
a[0] := 1; a[n_] := (-1)^n*2^(n + 1)/(n*(n + 1)^2)*Sum[(-1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k , 1, n}]; Table[a[n], {n, 0, 25}] (* Detlef Meya, May 29 2024 *)
  • Python
    def A368709(n):
    if n == 0: return 1
    return sum((-2)**k * v for k, v in enumerate(A359363Row(n))) // (-2)
print([A368709(n) for n in range(26)]) # Peter Luschny, Jan 04 2024
  • SageMath
    def A368709(n): return PolyA359363(n, -2) // (-2) if n > 0 else 1
print([A368709(n) for n in range(0, 26)])  # Peter Luschny, Jan 04 2024

Formula

a(n) = (-1/2)*B(n, -2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - Peter Luschny, Jan 04 2024
a(0) = 1, a(n) = (-1)^n*2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (-1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - Detlef Meya, May 29 2024
Showing 1-5 of 5 results.

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