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
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.
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.
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
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.
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.
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
-
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
-
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 *)
-
a(n) = binomial(2*n,n)-sum(j=0,n-1,binomial(2*j,j)); \\ Indranil Ghosh, Mar 03 2017
-
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
-
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
-
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
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