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 A279565 Number of length n inversion sequences avoiding the patterns 100, 110, 120, 201, and 210. 23
 1, 1, 2, 6, 21, 81, 332, 1420, 6266, 28318, 130412, 609808, 2887582, 13818590, 66726628, 324713196, 1590853485, 7840315329, 38843186366, 193342353214, 966409013021, 4848846341569, 24412146213116, 123290812268404, 624448756434476, 3171046361310556 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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_k. This is the same as the set of length n inversion sequences avoiding 100, 110, 120, 201, and 210. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1372 Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016. FORMULA G.f.: 3/(4-4*sin(asin((27*x+11)/16)/3)). - Vladimir Kruchinin, Mar 25 2019 a(n) = (1/n)*Sum_{m=1..n} m*Sum_{k=0..n-m} C(k,n-m-k)*C(n+k-1,k), n>0, a(0)=1. - Vladimir Kruchinin, Mar 26 2019 EXAMPLE The length 4 inversion sequences avoiding (100, 110, 120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123. MAPLE a:= proc(n) option remember; `if`(n<3, n!,       ((n-1)*(17*n-28)*a(n-1) +(49*n^2-185*n+196)*a(n-2)        +(3*(3*n-7))*(3*n-8)*a(n-3)) / (5*n*(n-1)))     end: seq(a(n), n=0..30);  # Alois P. Heinz, Feb 22 2017 MATHEMATICA a[n_] := a[n] = If[n < 3, n!, (((n - 1)*(17*n - 28)*a[n-1] + (49*n^2 - 185*n + 196)*a[n-2] + (3*(3*n - 7))*(3*n - 8)*a[n-3]) / (5*n*(n - 1)))]; Array[a, 30, 0] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *) Join[{1}, Table[(1/n)*Sum[m*Sum[Binomial[k, n-m-k]*Binomial[n+k-1, k], {k, 0, n-m}], {m, 1, n}], {n, 1, 30}]] (* G. C. Greubel, Mar 29 2019 *) PROG (Maxima) a(n):=if n=0 then 1 else sum(m*sum(binomial(k, n-m-k)*binomial(n+k-1, k), k, 0, n-m), m, 1, n)/n /* Vladimir Kruchinin, Mar 26 2019 */ (PARI) my(x='x+O('x^30)); Vec(round(3/(4-4*sin(asin((27*x+11)/16)/3)))) \\ G. C. Greubel, Mar 29 2019 (MAGMA) I:=[6, 21, 81]; [1, 1, 2] cat [n le 3 select I[n] else ( (n+1)*(17*n+6)*Self(n-1) +(49*n^2+11*n+22)*Self(n-2) +3*(3*n-1)*(3*n-2)*Self(n-3) )/(5*(n+2)*(n+1)) : n in [1..30]]; // G. C. Greubel, Mar 29 2019 (Sage) [1] +[(1/n)*(sum(sum(k*binomial(j, n-k-j)*binomial(n+j-1, j) for j in (0..n-k)) for k in (1..n))) for n in (1..30)] # G. C. Greubel, Mar 29 2019 CROSSREFS Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573. Sequence in context: A148494 A150212 A150213 * A150214 A150215 A328434 Adjacent sequences:  A279562 A279563 A279564 * A279566 A279567 A279568 KEYWORD nonn AUTHOR Megan A. Martinez, Feb 09 2017 EXTENSIONS a(10)-a(25) from Alois P. Heinz, Feb 22 2017 STATUS approved

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Last modified February 27 11:47 EST 2020. Contains 332305 sequences. (Running on oeis4.)