A071155 -id:A071155 - cp's OEIS frontend

A071157 The zero-free, right-to-left factorial walk encoding for each rooted plane tree encoded by A014486. Sequence A071155 shown with factorial expansion (A007623).

0, 1, 11, 21, 111, 211, 121, 221, 321, 1111, 2111, 1211, 2211, 3211, 1121, 2121, 1221, 2221, 3221, 1321, 2321, 3321, 4321, 11111, 21111, 12111, 22111, 32111, 11211, 21211, 12211, 22211, 32211, 13211, 23211, 33211, 43211, 11121, 21121, 12121

Offset: 0

Antti Karttunen, May 14 2002

Apart from the initial term (0, which encodes the null tree), if we scan the digits from the right (the least significant digit which is always 1) to the left (the most significant), then each successive digit to the left is at most one greater than the previous and never less than one.

Note: this finite decimal representation works only up to the 23712nd term, as the 23713rd such walk is already (10,9,8,7,6,5,4,3,2,1). The sequence A071158 shows the initial portion of this sequence sorted.

C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou et al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
A. Karttunen, Gatomorphisms and other excursions amidst the plane trees and parenthesizations (Includes the complete Scheme program for computing this sequence)

Corresponding Łukasiewicz words: A071153.

Essentially the same as A071159 but with digits reversed.

a(n) = A007623(A071155(n)).

A085200 Inverse function of N -> N injection A071155.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 0, 0, 4, 0, 6, 0, 0, 0, 5, 0, 7, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 14, 0, 0, 0, 11, 0, 16, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 15, 0, 0, 0, 12, 0, 17, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 18, 0, 0, 0, 0, 0, 21, 0, 0, 0

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

a(0)=0 because A071155(0)=0, but a(n) = 0 also for those n which do not occur as values of A071155. All positive natural numbers occur here once.

a(A071155(n)) = n for all n. Cf. A080300.

A071153 Łukasiewicz word for each rooted plane tree (interpretation e in Stanley's exercise 19) encoded by A014486 (or A063171), with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0.

Original entry on oeis.org

0, 1, 20, 11, 300, 201, 210, 120, 111, 4000, 3001, 3010, 2020, 2011, 3100, 2101, 2200, 1300, 1201, 2110, 1210, 1120, 1111, 50000, 40001, 40010, 30020, 30011, 40100, 30101, 30200, 20300, 20201, 30110, 20210, 20120, 20111, 41000, 31001, 31010

Offset: 0

Antti Karttunen, May 14 2002

Note: this finite decimal representation works only up to the 6917th term, as the 6918th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071154 shows the initial portion of this sequence sorted.

			The 11th term of A063171 is 10110010, corresponding to parenthesization ()(())(), thus its Łukasiewicz word is 3010. The 18th term of A063171 is 11011000, corresponding to parenthesization (()(())), thus its Łukasiewicz word is 1201. I.e., in the latter example there is one list on the top-level, which in turn contains two sublists, of which the first is zero elements long and the second is a sublist containing one empty sublist (the last zero is omitted).

A. Karttunen, Gatomorphisms and other excursions amidst the plane trees and parenthesizations (Includes the complete Scheme program for computing this sequence)
R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Exercises on Catalan and Related Numbers
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to parenthesizing

For n >= 1, the number of zeros in the term a(n) is given by A057514(n)-1.
The first digit of each term is given by A057515.
Cf. A014486, A059984, A059985, A071152, A071154.
Corresponding factorial walk encoding: A071155 (A071157, A071159).
a(n) = A079436(n)/10.

A071156 Apart from the initial term (0), lists all integers whose factorial expansion ends with 1 (i.e., are odd numbers), do not contain a digit zero and each successive digit to the left is at most one greater than the preceding digit.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 23, 33, 35, 39, 41, 47, 57, 59, 63, 65, 71, 87, 89, 95, 119, 153, 155, 159, 161, 167, 177, 179, 183, 185, 191, 207, 209, 215, 239, 273, 275, 279, 281, 287, 297, 299, 303, 305, 311, 327, 329, 335, 359, 417, 419, 423, 425, 431, 447, 449

Offset: 0

Antti Karttunen, May 14 2002

a(n) = A085198(A014486(n)) = A071155(A057164(n)). Catalan numbers A000108(n) gives the number of terms whose factorial expansion contain n digits.

0 is included by considering it to have the empty string as its factorial base representation. - Franklin T. Adams-Watters, Jun 28 2006

The beginning of this sequence expanded in the factorial number system: A071158. Inverse function: A085199. First differences: A085191.

Cf. A000108 (row lengths), A071155, A120696.

A085201 Array A(x,y): Position of the concatenation of binary strings A014486(x) & A014486(y) in A014486, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 9, 5, 4, 5, 9, 14, 10, 9, 5, 6, 11, 23, 15, 23, 10, 6, 7, 14, 28, 24, 37, 24, 11, 7, 8, 16, 37, 29, 65, 38, 25, 12, 8, 9, 19, 42, 38, 79, 66, 39, 26, 13, 9, 10, 23, 51, 43, 107, 80, 67, 40, 27, 23, 10, 11, 25, 65, 52, 121, 108, 81, 68, 41, 65, 24, 11

Offset: 0

Antti Karttunen, Jun 23 2003

This table is induced by the 2-ary form of the list-function 'append' present in the programming languages like Lisp, Scheme and Prolog.

Index entries for the sequences induced by list functions of Lisp

Transpose: A085202. Variant: A085203. Row 1: A085223, Column 1: A072795.

Cf. also A014486, A080300, A025581, A002262.

a(0, y)=y, a(x, y) = A072764bi(A072771(x), a(A072772(x), y))

a(x, y) = A080300(A085207bi(A014486(x), A014486(y))) = A085200(A085215bi(A071155(y), A071155(x)))

A085203 Array A(x,y): Position of the totally balanced binary string obtained by concatenating the binary strings A014486(x) & A014486(y) in such a way that the latter is inserted after the least significant 1-bit of the former, followed by the remaining 0-bits, if any. Listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 7, 3, 4, 8, 12, 8, 4, 5, 10, 21, 13, 17, 5, 6, 13, 26, 22, 31, 18, 6, 7, 15, 35, 27, 58, 32, 20, 7, 8, 18, 40, 36, 73, 59, 34, 21, 8, 9, 22, 49, 41, 100, 74, 62, 35, 22, 9, 10, 24, 63, 50, 115, 101, 76, 63, 36, 45, 10, 11, 27, 68, 64, 142, 116, 104, 77, 64, 87, 46

Offset: 0

Antti Karttunen, Jun 23 2003

This table is induced by the list-function 'app-to-xrt' whose Scheme-definition is given below, in the same way as A085201 is induced by the ordinary 'append'-function.

Transpose: A085204. Variant: A085201. Row 1: A085225, Column 1: A057548.

Cf. also A014486, A080300, A025581, A002262.

a(0, y) = y, a(x, y) = A057548(a(A072771(x), y)) if A072772(x)=0, otherwise A072764bi(A072771(x), a(A072772(x), y)).

a(x, y) = A080300(A085211bi(A014486(x), A014486(y))) = A085200(A085219bi(A071155(y), A071155(x))).

A236855 a(n) is the sum of digits in A239903(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 5, 6, 7, 8, 4, 5, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 11, 3, 4

Offset: 0

Antti Karttunen, Apr 18 2014

			As the 0th Catalan String is empty, indicated by A239903(0)=0, a(0)=0.
As the 18th Catalan String is [1,0,1,2] (A239903(18)=1012), a(18) = 1+0+1+2 = 4.
Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 55, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.

Antti Karttunen, Table of n, a(n) for n = 0..16796

Cf. A239903, A014420, A237449, A236859, A009766, A071155, A081291, A034968, A060130.

A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
A236855list[6] (* Generates C(6) terms *) (* Paolo Xausa, Feb 19 2024 *)

(define (A236855 n) (apply + (A239903raw n)))
(define (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (- (A081288 n) 1)) (col (- (A081288 n) 2)) (srow (- (A081288 n) 2)) (catstring (list 0))) (cond ((or (zero? row) (negative? col)) (reverse! (cdr catstring))) ((> (A009766tr row col) n) (loop n srow (- col 1) (- srow 1) (cons 0 catstring))) (else (loop (- n (A009766tr row col)) (+ row 1) col srow (cons (+ 1 (car catstring)) (cdr catstring))))))))
;; Alternative definition:
(define (A236855 n) (let ((x (A071155 (A081291 n)))) (- (A034968 x) (A060130 x))))

a(n) = A034968(x) - A060130(x), where x = A071155(A081291(n)).
For up to n = A000108(11)-2 = 58784, a(n) = A007953(A239903(n)).
Catalan numbers, A000108, give the positions of ones, and the n-th triangular number occurs for the first time at the position immediately before that, i.e., a(A001453(n)) = A000217(n-1).
For each n, a(n) >= A000217(A236859(n)).

A120695 Set partitions reversed interpreted as factorial base numbers.

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 11, 17, 23, 33, 57, 39, 63, 87, 35, 59, 83, 41, 65, 89, 47, 71, 95, 119, 153, 273, 177, 297, 417, 159, 279, 399, 183, 303, 423, 207, 327, 447, 567, 155, 275, 395, 179, 299, 419, 203, 323, 443, 563, 161, 281, 401, 185, 305, 425, 209, 329, 449

Offset: 0

Franklin T. Adams-Watters, Jun 28 2006

			The third set partition is {1,2}, 21 in base factorial is 5, so a(3) = 5.
Triangle begins:
   0;
   1;
   3,  5;
   9, 15, 11, 17, 23;
  33, 57, 39, 63, 87, 35, 59, 83, 41, 65, 89, 47, 71, 95, 119;

Alois P. Heinz, Rows n = 0..9, flattened

Cf. A000110 (row lengths), A120698, A120696 (sorted), A120697, A071155.

Column k=0 gives A007489.

b:= proc(n, m, t) option remember; `if`(n=0, [0], [seq(map(
       x-> x+j*t!, b(n-1, max(m, j), t+1))[], j=1..m+1)])
    end:
T:= n-> b(n, 0, 1)[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Apr 04 2016

cp's OEIS Frontend

A071157 The zero-free, right-to-left factorial walk encoding for each rooted plane tree encoded by A014486. Sequence A071155 shown with factorial expansion (A007623).

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A085200 Inverse function of N -> N injection A071155.

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A071153 Łukasiewicz word for each rooted plane tree (interpretation e in Stanley's exercise 19) encoded by A014486 (or A063171), with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0.

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A071156 Apart from the initial term (0), lists all integers whose factorial expansion ends with 1 (i.e., are odd numbers), do not contain a digit zero and each successive digit to the left is at most one greater than the preceding digit.

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A085201 Array A(x,y): Position of the concatenation of binary strings A014486(x) & A014486(y) in A014486, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

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A236855 a(n) is the sum of digits in A239903(n).

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Mathematica

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A120695 Set partitions reversed interpreted as factorial base numbers.

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Maple