OFFSET

0,8

COMMENTS

T(n,k) is also the number of length n bit strings beginning with 0 having k singletons. Example: T(4,2)=3 because we have 0010, 0100 and 0110. - Emeric Deutsch, Sep 21 2008

The cyclic version of this array is A320341(n,k), which counts the (unmarked) cyclic compositions of n with exactly k parts equal to 1, with a minor exception for k=0. The sequence (A320341(n, k=0) - 1: n >= 1) counts the (unmarked) cyclic compositions of n with no parts equal to 1. - Petros Hadjicostas, Jan 08 2019

Also the convolution triangle of Fibonacci(n-2). # Peter Luschny, Oct 07 2022

T(n,k) is the number of length n+1 bit strings beginning and ending with 0 having k length 2 substrings 00. This is equivalent to the compositions interpretation because each m part corresponds to a length m+1 bit string beginning with 0 and ending with the next 0 bit. Thus a substring 00 corresponds to a 1 part. Example: T(4,2)=3 because we have 00010 for 112, 00100 for 121 and 01000 for 211. - Michael Somos, Sep 24 2024

In the Baccherini et al. 2008 link on page 81: "Bloom[3] studies the number of singles in all the 2^n n-length bit strings, where a _single_ is any isolated 1 or 0, i.e., any run of length 1. Let R_{n,k} be the number of n-length bit strings beginning with 0 and having k singles." Here T(n,k) = R_{n,k}. This combinatorial interpretation is equivalent to my previous comment since a part of size k corresponds to run of k identical bits and also to a length k+1 bit string with 0s only at the beginning and end. - Michael Somos, Sep 25 2024

LINKS

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

D. Baccherini, D. Merlini and R. Sprugnoli, Level generating trees and proper Riordan arrays, Applicable Analysis and Discrete Mathematics, 2, 2008, 69-91 (see p. 83). [Emeric Deutsch, Sep 21 2008]

Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 10-11.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

FORMULA

G.f.: (1-z)/(1 - z - z^2 - tz + tz^2).

T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0)=1, T(1,0)=0. - Philippe Deléham, Nov 18 2009

If the triangle's columns are transposed into rows, then T(n,k) can be interpreted as the number of compositions of n+k having exactly k 1's. Then g.f.: ((1-x)/(1-x-x^2))^(k-1) and T(n,k) = T(n-2,k) + T(n-1,k) - T(n-1, k-1) + T(n, k-1). Also, Sum_{j=1..n} T(n-j, j) = 2^(n-1), the number of compositions of n. - Gregory L. Simay, Jun 28 2019

EXAMPLE

T(6,2) = 9 because we have (1,1,4), (1,4,1), (4,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1) and (2,2,1,1).

Triangle begins:

00: 1;

01: 0, 1;

02: 1, 0, 1;

03: 1, 2, 0, 1;

04: 2, 2, 3, 0, 1;

05: 3, 5, 3, 4, 0, 1;

06: 5, 8, 9, 4, 5, 0, 1;

07: 8, 15, 15, 14, 5, 6, 0, 1;

08: 13, 26, 31, 24, 20, 6, 7, 0, 1;

09: 21, 46, 57, 54, 35, 27, 7, 8, 0, 1;

10: 34, 80, 108, 104, 85, 48, 35, 8, 9, 0, 1;

11: 55, 139, 199, 209, 170, 125, 63, 44, 9, 10, 0, 1;

12: 89, 240, 366, 404, 360, 258, 175, 80, 54, 10, 11, 0, 1;

13: 144, 413, 666, 780, 725, 573, 371, 236, 99, 65, 11, 12, 0, 1;

...

MAPLE

G:=(1-z)/(1-z-z^2-t*z+t*z^2): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 13 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form

# second Maple program:

T:= proc(n) option remember; local j; if n=0 then 1

else []; for j to n do zip((x, y)-> x+y, %,

[`if`(j=1, 0, [][]), T(n-j)], 0) od; %[] fi

end:

seq(T(n), n=0..20); # Alois P. Heinz, Nov 05 2012

# Uses function PMatrix from A357368. Adds a row above and a column to the left.

PMatrix(10, n -> combinat:-fibonacci(n-2)); # Peter Luschny, Oct 07 2022

MATHEMATICA

nn = 10; a = x/(1 - x) - x + y x;

CoefficientList[CoefficientList[Series[1/(1 - a), {x, 0, nn}], x], y] // Flatten (* Geoffrey Critzer, Dec 23 2011 *)

T[ n_, k_] := Which[k<0 || k>n, 0, n<2, Boole[n==k], True, T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 24 2024 *)

PROG

(PARI) {T(n, k) = if(k<0 || k>n, 0, n<2, n==k, T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 24 2024 */

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 07 2005

STATUS

approved