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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A279558 Number of length n inversion sequences avoiding the patterns 010, 120, and 210.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 200, 830, 3654, 16869, 80963, 401300, 2043610, 10649335, 56604706, 306101789, 1680515427
Offset: 0

Views

Author

Megan A. Martinez, Jan 17 2017

Keywords

Comments

A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_j > e_k and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 010, 120, and 210.

Examples

			The length 4 inversion sequences avoiding (010, 120, 210) are 0000, 0001, 0002, 0003, 0011, 0012, 0013, 0021, 0022, 0023, 0111, 0112, 0113, 0122, 0123.

Links

Crossrefs

Cf. A000108, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Extensions

a(10)-a(12) from Alois P. Heinz, Feb 24 2017
a(13)-a(16) from Bert Dobbelaere, Dec 30 2018

A279572 Number of length n inversion sequences avoiding the patterns 120, 201, and 210.

Original entry on oeis.org

1, 1, 2, 6, 23, 101, 484, 2468, 13166, 72630, 411076, 2374188, 13938018, 82932254, 499031324, 3031610924, 18568429963, 114541486785, 710973143614, 4437415155234, 27831038618735, 175318861863701, 1108762012137252, 7037137177329268, 44808588430903068
Offset: 0

Views

Author

Megan A. Martinez, Feb 21 2017

Keywords

Comments

A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_j <> e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 120, 201, and 210.

Examples

			The length 4 inversion sequences avoiding (120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0121, 0122, 0123.

Links

Crossrefs

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279573.

Extensions

a(12)-a(15) from Bert Dobbelaere, Dec 30 2018
a(16)-a(24) from Toufik Mansour et al. added by Stefano Spezia, Jan 20 2024

A114277 Sum of the lengths of the second ascents in all Dyck paths of semilength n+2.

Original entry on oeis.org

1, 5, 19, 67, 232, 804, 2806, 9878, 35072, 125512, 452388, 1641028, 5986993, 21954973, 80884423, 299233543, 1111219333, 4140813373, 15478839553, 58028869153, 218123355523, 821908275547, 3104046382351, 11747506651599
Offset: 0

Views

Author

Emeric Deutsch, Nov 20 2005

Keywords

Comments

Also number of Dyck paths of semilength n+4 having length of second ascent equal to three. Example: a(1)=5 because we have UD(UUU)DUDDD, UD(UUU)DDUDD, UD(UUU)DDDUD, UUD(UUU)DDDD and UUDD(UUU)DDD (second ascents shown between parentheses). Partial sums of A002057. Column 3 of A114276. a(n)=absolute value of A104496(n+3).
Also number of Dyck paths of semilength n+3 that do not start with a pyramid (a pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis; here U=(1,1) and D=(1,-1); this definition differs from the one in A091866). Equivalently, a(n)=A127156(n+3,0). Example: a(1)=5 because we have UUDUDDUD, UUDUDUDD, UUUDUDDD, UUDUUDDD and UUUDDUDD. - Emeric Deutsch, Feb 27 2007

Examples

			a(3)=5 because the total length of the second ascents in UD(U)DUD, UD(UU)DD, UUDD(U)D, UUD(U)DD and UUUDDD (shown between parentheses) is 5.

Links

Crossrefs

Cf. A002057, A114276, A104496, A127156, A279557.
Cf. A014137 (n=1), A014138 (n=2), A001453 (n=3), this sequence (n=4), A143955 (n=5), A323224 (array).

Programs

  • Maple
    a:=n->4*sum(binomial(2*j+3,j)/(j+4),j=0..n): seq(a(n),n=0..28);
  • Mathematica
    Table[4*Sum[Binomial[2j+3,j]/(j+4),{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 19 2012 *)
  • Python
    from functools import cache
@cache
def B(n, k):
    if n <= 0 or k <= 0: return 0
    if n == k: return 1
    return B(n - 1, k) + B(n, k - 1)
def A114277(n): return B(n + 5, n + 1)
print([A114277(n) for n in range(24)]) # Peter Luschny, May 16 2022

Formula

a(n) = 4*Sum_{j=0..n} binomial(2*j+3, j)/(j+4).
G.f.: C^4/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) = c(n+3) - (c(0) + c(1) + ... + c(n+2)), where c(k)=binomial(2k,k)/(k+1) is a Catalan number (A000108). - Emeric Deutsch, Feb 27 2007
D-finite with recurrence: n*(n+4)*a(n) = (5*n^2 + 14*n + 6)*a(n-1) - 2*(n+1)*(2*n+3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+7)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
a(n) = exp((2*i*Pi)/3)-4*binomial(2*n+5,n+1)*hypergeom([1,3+n,n+7/2],[n+2,n+6],4)/ (n+5). - Peter Luschny, Feb 26 2017
a(n-1) = Sum_{i+j+k+lA000108 Catalan number. - Yuchun Ji, Jan 10 2019

Extensions

More terms from Emeric Deutsch, Feb 27 2007

A281593 a(n) = b(n) - Sum_{j=0..n-1} b(j) with b(n) = binomial(2*n, n).

Original entry on oeis.org

1, 1, 3, 11, 41, 153, 573, 2157, 8163, 31043, 118559, 454479, 1747771, 6740059, 26055459, 100939779, 391785129, 1523230569, 5931153429, 23126146629, 90282147849, 352846964649, 1380430179489, 5405662979649, 21186405207549, 83101804279101, 326199124351701
Offset: 0

Views

Author

Peter Luschny, Feb 25 2017

Keywords

Links

Crossrefs

A279561(n) = (a(n)+1)/2.
A057552(n) = (a(n+2)-1)/2.
A162551(n) = a(n+1)-a(n).
Cf. A000984, A006134, A279557.

Programs

  • Maple
    b := n -> binomial(2*n, n): s := n -> add(b(j), j=0..n):
a := n -> b(n) - s(n-1): seq(a(n), n=0..26);
# second program:
A281593 := series(exp(2*x)*BesselI(0, 2*x) - exp(x)*int(BesselI(0, 2*x)*exp(x), x), x = 0, 27): seq(n!*coeff(A281593, x, n), n=0..26); # Mélika Tebni, Feb 27 2024
  • Mathematica
    a[n_] = Binomial[2n,n](1+Hypergeometric2F1[1,n+1/2,n+1,4])+I/Sqrt[3];
Table[Simplify[a[n]],{n,0,17}]
CoefficientList[Series[(2x -1)/((x -1) Sqrt[(1 -4x)]), {x, 0, 26}], x] (* Robert G. Wilson v, Feb 25 2017 *)
a[0]=1; a[n_]:=a[n-1] + 2*(n-1)*CatalanNumber[n-1];Table[a[n],{n,0,26}] (* Indranil Ghosh, Mar 03 2017 *)
  • PARI
    a(n) = binomial(2*n,n)-sum(j=0,n-1,binomial(2*j,j)); \\ Indranil Ghosh, Mar 03 2017
  • PARI
    c(n) = binomial(2*n,n)/(n+1);
a(n) = if(n==0,1,a(n-1) + 2*(n-1)*c(n-1)); \\ Indranil Ghosh, Mar 03 2017
  • Python
    import math
def C(n,r): return f(n)/f(r)/f(n-r)
def A281593(n):
    s=0
    for j in range(0,n):
        s+=C(2*j,j)
    return C(2*n,n)-s # Indranil Ghosh, Mar 03 2017
  • Sage
    def A():
    a = b = c = 1
    yield 1
    while True:
        yield a
        c = (c * (4 * b - 2)) // (b + 1)
        a += 2 * b * c
        b += 1
a = A(); print([next(a) for  in (0..25)]) # _Peter Luschny, Feb 25 2017

Formula

a(n) = [x^n] (2*x-1)/(sqrt(1-4*x)*(x-1)).
a(n) = binomial(2*n,n)*(1+hypergeom([1,n+1/2],[n+1],4))+I/sqrt(3).
a(n+1) = a(n) + 2*n*Catalan(n).
a(n) ~ (4/3)*4^n/sqrt((8*n+2)*Pi/2).
D-finite with recurrence n*a(n) +(-7*n+6)*a(n-1) +2*(7*n-13)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jul 27 2022
E.g.f.: exp(2*x)*BesselI(0,2*x) - exp(x)*integral( BesselI(0,2*x)*exp(x) ) dx. - Mélika Tebni, Feb 27 2024
Previous Showing 21-24 of 24 results.

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