A027658 -id:A027658 - cp's OEIS frontend

A144903 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of x/((1-x-x^3)*(1-x)^(k-1)).

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 3, 2, 1, 0, 1, 4, 6, 5, 3, 1, 0, 1, 5, 10, 11, 8, 4, 2, 0, 1, 6, 15, 21, 19, 12, 6, 3, 0, 1, 7, 21, 36, 40, 31, 18, 9, 4, 0, 1, 8, 28, 57, 76, 71, 49, 27, 13, 6, 0, 1, 9, 36, 85, 133, 147, 120, 76, 40, 19, 9, 0, 1, 10, 45, 121, 218, 280, 267, 196, 116, 59, 28, 13

Offset: 0

Alois P. Heinz, Sep 24 2008

			Square array (A(n,k)) begins:
  0, 0,  0,  0,  0,   0,   0 ... A000004;
  1, 1,  1,  1,  1,   1,   1 ... A000012;
  0, 1,  2,  3,  4,   5,   6 ... A001477;
  0, 1,  3,  6, 10,  15,  21 ... A000217;
  1, 2,  5, 11, 21,  36,  57 ... A050407;
  1, 3,  8, 19, 40,  76, 133 ... ;
  1, 4, 12, 31, 71, 147, 200 ... A027658;
Antidiagonal triangle (T(n,k)) begins as:
  0;
  0,  1;
  0,  1,  0;
  0,  1,  1,  0;
  0,  1,  2,  1,  1;
  0,  1,  3,  3,  2,  1;
  0,  1,  4,  6,  5,  3,  1;
  0,  1,  5, 10, 11,  8,  4,  2;
  0,  1,  6, 15, 21, 19, 12,  6,  3;

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows 0-4, 6 give: A000004, A000012, A001477, A000217, A050407(n+3), A027658.
Columns 0-9 give: A078012 and A135851(n+2), A078012(n+2) and A135851(n+4), A077868(n-1) for n>0, A050228(n-1) for n>0, A226405, A144898, A144899, A144900, A144901, A144902.
Main diagonal gives: A144904.
Cf. A000930.

A000930:= func< n | (&+[Binomial(n-2*j,j): j in [0..Floor(n/3)]]) >;
A144903:= func< n,k | k eq 0 select 0 else (&+[Binomial(n-k+j-2,j)*A000930(k-j-1) : j in [0..k-1]]) >;
[A144903(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 01 2022

A:= proc(n,k) coeftayl (x/ (1-x-x^3)/ (1-x)^(k-1), x=0, n) end:
seq(seq(A(n, d-n), n=0..d), d=0..13);

(* First program *)
a[n_, k_] := SeriesCoefficient[x/((1-x-x^3)*(1-x)^(k-1)), {x, 0, n}];
Table[a[n-k, k], {n,0,12}, {k,n,0,-1}]//Flatten (* Jean-François Alcover, Jan 15 2014 *)
(* Second Program *)
A000930[n_]:= A000930[n]= Sum[Binomial[n-2*j,j], {j,0,Floor[n/3]}];
T[n_, k_]:= T[n, k]= If[k==0, 0, Sum[Binomial[n-k+j-2,j]*A000930[k-j-1], {j,0,k- 1}]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2022 *)

def A000930(n): return sum(binomial(n-2*j,j) for j in (0..(n//3)))
def A144903(n,k):
    if (k==0): return 0
    else: return sum(binomial(n-k+j-2,j)*A000930(k-j-1) for j in (0..k-1))
flatten([[A144903(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Aug 01 2022

G.f. of column k: x/((1-x-x^3)*(1-x)^(k-1)).
A(n, n) = A144904(n).
From G. C. Greubel, Aug 01 2022: (Start)
A(n, k) = Sum_{j=0..n-1} binomial(k+j-2, j)*A000930(n-j-1), with A(0, k) = 0.
T(n, k) = Sum_{j=0..k-1} binomial(n-k-j-2, j)*A000930(k-j-1), with T(n, 0) = 0.
T(2*n, n) = A144904(n). (End)

A179257 Number of permutations of length n which avoid the patterns 321 and 1324.

Original entry on oeis.org

1, 1, 2, 5, 13, 32, 72, 148, 281, 499, 838, 1343, 2069, 3082, 4460, 6294, 8689, 11765, 15658, 20521, 26525, 33860, 42736, 53384, 66057, 81031, 98606, 119107, 142885, 170318, 201812, 237802, 278753, 325161, 377554, 436493, 502573, 576424, 658712, 750140, 851449

Offset: 0

Vincent Vatter, Jul 05 2010

			There are 13 permutations of length 4 which avoid these two patterns, so a(4)=13.

M. D. Atkinson, Restricted permutations, Discrete Math., 195 (1999), 27-38.
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
J. West, Generating trees and forbidden subsequences, Discrete Math., 157 (1996), 363-374.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A116699, A116701, A116702, A088921, A005183, A116703, A001519.

LinearRecurrence[{6,-15,20,-15,6,-1},{1,1,2,5,13,32},50] (* Harvey P. Dale, May 19 2024 *)

a(n) = 1+binomial(n,2)+binomial(n+2,5).
G.f.: 1-x*(x^5-4*x^4+7*x^3-8*x^2+4*x-1)/(x-1)^6. - Colin Barker, Aug 02 2012
a(n) = 1+A027658(n-2). - R. J. Mathar, Aug 19 2022

a(0)=1 prepended by Alois P. Heinz, Jul 05 2018

cp's OEIS Frontend

A144903 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of x/((1-x-x^3)*(1-x)^(k-1)).

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