cp's OEIS Frontend

There are 2 versions of Euler's triangle:

* A008292 Classic version of Euler's triangle used by Comtet (1974).

* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).

Euler's triangle rows and columns indexing conventions:

* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)

* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0. (Graham et al.)

Number of Dyck paths of semilength n having exactly one long ascent (i.e., ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004

Number of permutations of {1,2,...,n} with exactly one descent (i.e., permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g., a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312.

a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079). - Graeme McRae, Jun 07 2006

Partial sum of main diagonal of A125127. - Jonathan Vos Post, Nov 22 2006

Number of partitions of an n-set having exactly one block of size > 1. Example: a(4)=11 because, if the partitioned set is {1,2,3,4}, then we have 1234, 123|4, 124|3, 134|2, 1|234, 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34. - Emeric Deutsch, Oct 28 2006

k divides a(k+1) for k in A014741. - Alexander Adamchuk, Nov 03 2006

(Number of permutations avoiding patterns 321, 2413, 3412, 21534) minus one. - Jean-Luc Baril, Nov 01 2007, Mar 21 2008

The chromatic invariant of the prism graph P_n for n >= 3. - Jonathan Vos Post, Aug 29 2008

Decimal integer corresponding to the result of XORing the binary representation of 2^n - 1 and the binary representation of n with leading zeros. This sequence and a few others are syntactically similar. For n > 0, let D(n) denote the decimal integer corresponding to the binary number having n consecutive 1's. Then D(n).OP.n represents the n-th term of a sequence when .OP. stands for a binary operator such as '+', '-', '*', 'quotentof', 'mod', 'choose'. We then get the various sequences A136556, A082495, A082482, A066524, A000295, A052944. Another syntactically similar sequence results when we take the n-th term as f(D(n)).OP.f(n). For example if f='factorial' and .OP.='/', we get (A136556)(A000295) ; if f='squaring' and .OP.='-', we get (A000295)(A052944). - K.V.Iyer, Mar 30 2009

Chromatic invariant of the prism graph Y_n.

Number of labelings of a full binary tree of height n-1, such that each path from root to any leaf contains each label from {1,2,...,n-1} exactly once. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009

Also number of nontrivial equivalence classes generated by the weak associative law X((YZ)T)=(X(YZ))T on words with n open and n closed parentheses. Also the number of join (resp. meet)-irreducible elements in the pruning-grafting lattice of binary trees with n leaves. - Jean Pallo, Jan 08 2010

Nonzero terms of this sequence can be found from the row sums of the third sub-triangle extracted from Pascal's triangle as indicated below by braces:

1;

1, 1;

{1}, 2, 1;

{1, 3}, 3, 1;

{1, 4, 6}, 4, 1;

{1, 5, 10, 10}, 5, 1;

{1, 6, 15, 20, 15}, 6, 1;

... - L. Edson Jeffery, Dec 28 2011

For integers a, b, denote by a<+>b the least c >= a, such that the Hamming distance D(a,c) = b (note that, generally speaking, a<+>b differs from b<+>a). Then for n >= 3, a(n) = n<+>n. This has a simple explanation: for n >= 3 in binary we have a(n) = (2^n-1)-n = "anti n". - Vladimir Shevelev, Feb 14 2012

a(n) is the number of binary sequences of length n having at least one pair 01. - Branko Curgus, May 23 2012

Nonzero terms are those integers k for which there exists a perfect (Hamming) error-correcting code. - L. Edson Jeffery, Nov 28 2012

a(n) is the number of length n binary words constructed in the following manner: Select two positions in which to place the first two 0's of the word. Fill in all (possibly none) of the positions before the second 0 with 1's and then complete the word with an arbitrary string of 0's or 1's. So a(n) = Sum_{k=2..n} (k-1)*2^(n-k). - Geoffrey Critzer, Dec 12 2013

Without first 0: a(n)/2^n equals Sum_{k=0..n} k/2^k. For example: a(5)=57, 57/32 = 0/1 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32. - Bob Selcoe, Feb 25 2014

The first barycentric coordinate of the centroid of the first n rows of Pascal's triangle, assuming the numbers are weights, is A000295(n+1)/A000337(n). See attached figure. - César Eliud Lozada, Nov 14 2014

Starting (0, 1, 4, 11, ...), this is the binomial transform of (0, 1, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015

Also the number of (non-null) connected induced subgraphs in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Aug 27 2017

a(n) is the number of swaps needed in the worst case to transform a binary tree with n full levels into a heap, using (bottom-up) heapify. - Rudy van Vliet, Sep 19 2017

The utility of large networks, particularly social networks, with n participants is given by the terms a(n) of this sequence. This assertion is known as Reed's Law, see the Wikipedia link. - Johannes W. Meijer, Jun 03 2019

a(n-1) is the number of subsets of {1..n} in which the largest element of the set exceeds by at least 2 the next largest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,2,4}, {1,2,5}, {1,3,5}, {2,3,5}, {1,2,3,5}. - Enrique Navarrete, Apr 08 2020

a(n-1) is also the number of subsets of {1..n} in which the second smallest element of the set exceeds by at least 2 the smallest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,4,5}, {1,3,4,5}. - Enrique Navarrete, Apr 09 2020

a(n+1) is the sum of the smallest elements of all subsets of {1..n}. For example, for n=3, a(4)=11; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of smallest elements is 11. - Enrique Navarrete, Aug 20 2020

Number of subsets of an n-set that have more than one element. - Eric M. Schmidt, Mar 13 2021

Number of individual bets in a "full cover" bet on n-1 horses, dogs, etc. in different races. Each horse, etc. can be bet on or not, giving 2^n bets. But, by convention, singles (a bet on only one race) are not included, reducing the total number bets by n. It is also impossible to bet on no horses at all, reducing the number of bets by another 1. A full cover on 4 horses, dogs, etc. is therefore 6 doubles, 4 trebles and 1 four-horse etc. accumulator. In British betting, such a bet on 4 horses etc. is a Yankee; on 5, a super-Yankee. - Paul Duckett, Nov 17 2021

From Enrique Navarrete, May 25 2022: (Start)

Number of binary sequences of length n with at least two 1's.

a(n-1) is the number of ways to choose an odd number of elements greater than or equal to 3 out of n elements.

a(n+1) is the number of ways to split [n] = {1,2,...,n} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n} and then select a subset from the first interval (2^i choices, 0 <= i <= n), and one block/cell (i.e., subinterval) from the second interval (n-i choices, 0 <= i <= n).

(End)

Number of possible conjunctions in a system of n planets; for example, there can be 0 conjunctions with one planet, one with two planets, four with three planets (three pairs of planets plus one with all three) and so on. - Wendy Appleby, Jan 02 2023

Largest exponent m such that 2^m divides (2^n-1)!. - Franz Vrabec, Aug 18 2023

It seems that a(n-1) is the number of odd r with 0 < r < 2^n for which there exist u,v,w in the x-independent beginning of the Collatz trajectory of 2^n x + r with u+v = w+1, as detailed in the link "Collatz iteration and Euler numbers?". A better understanding of this might also give a formula for A374527. - Markus Sigg, Aug 02 2024

This sequence has a connection to consecutively halved positional voting (CHPV); see Mendenhall and Switkay. - Hal M. Switkay, Feb 25 2025

a(n) is the number of subsets of size 2 and more of an n-element set. Equivalently, a(n) is the number of (hyper)edges of size 2 and more in a complete hypergraph of n vertices. - Yigit Oktar, Apr 05 2025

The numbers (4, 20, 120, 840, 6720, ...) arise from the divisor values in the general formula a(n) = n*(n+1)*(n+2)*(n+3)* ... *(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) (which covers the following sequences: A000578, A000537, A024166, A101094, A101097, A101102). - Alexander R. Povolotsky, May 17 2008

a(n) is also the number of decreasing 3-cycles in the decomposition of permutations as product of disjoint cycles, a(3)=1, a(4)=4, a(5)=20. - Wenjin Woan, Dec 21 2008

Equals eigensequence of triangle A130128 reflected. - Gary W. Adamson, Dec 23 2008

a(n) is the number of n-permutations having 1, 2, and 3 in three distinct cycles. - Geoffrey Critzer, Apr 26 2009

From Johannes W. Meijer, Oct 20 2009: (Start)

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=4) ~ exp(-x)/x*(1 - 4/x + 20/x^2 - 120/x^3 + 840/x^4 - 6720/x^5 + 60480/x^6 - 604800/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information.

(End)

Essentially a duplicate of A000295: a(n) = A000295(n+1).

Partial sums of main diagonal of array A125127 = L(k,n): k-step Lucas numbers, read by antidiagonals.

Equals row sums of triangle A130128. - Gary W. Adamson, May 11 2007

Row sums of triangle A130330 which is composed of (1,3,7,15,...) in every column, thus: row sums of (1; 3,1; 7,3,1; ...). - Gary W. Adamson, May 24 2007

Row sums of triangle A131768. - Gary W. Adamson, Jul 13 2007

Convolution A130321 * (1, 2, 3, ...). Binomial transform of (1, 3, 4, 4, 4, ...). - Gary W. Adamson, Jul 27 2007

Row sums of triangle A131816. - Gary W. Adamson, Jul 30 2007

A000975 convolved with [1, 2, 2, 2, ...]. - Gary W. Adamson, Jun 02 2009

The eigensequence of a triangle with the triangular series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

From Peter Munn, Apr 28 2025: (Start)

Each row in the Wythoff array, A035513, and this extended array satisfies the Fibonacci recurrence; that is each term after the first 2 is the sum of the preceding 2 terms.

We use F_i to denote the i-th Fibonacci term, A000045(i). In particular, we refer below to F_0 = 0, F_1 = 1 and F_2 = 1 several times. Note that to fully understand the description of the relationship between neighboring columns it is important to distinguish F_1 and F_2, although they have the same integer value. Similarly, the identity of an array term should be understood here as including its position in the array, not only its integer value.

The terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. With this map each term is the sum of its subset image. See the table in the examples.

Full description of the mapping with its relationship to A035513:

The (unextended) Wythoff array A035513 includes every positive integer exactly once. So, using the Zeckendorf representation (see link below), the array terms map 1:1 to nonempty finite subsets of the Fibonacci terms from F_2 onwards -- more precisely, onto those that do not include both F_i and F_{i+1} for any i. (Again, each array term is the sum of the Fibonacci numbers from the relevant subset.)

As shown in the Kimberling 1995 link, when we proceed from one term to the next in a row, the indices of the Fibonacci terms in the corresponding subset are incremented. When we proceed leftwards, the indices are decremented, with the subsets for the leftmost column being those that include F_2.

And when we add 2 columns on the left of the Wythoff array, the mapping continues to decrement the indices, so the corresponding extra subsets have F_0 (new leftmost column) or F_1 as their first Fibonacci term.

Thus the terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. The leftmost column is the nonnegative integers: if we were to remove F_0 (value 0) from the subset for an integer in this column, the subset would form the Zeckendorf representation of the integer, as subsets do in the unextended array.

(End)

Note the Zeckendorf representation of 0 is taken to be the empty sum.

The Wythoff array A035513 is the subtable formed by rows 3, 11, 16, 24, 32, ... (A035337). If, instead, we use rows 2, 7, 10, 15, 20, ... (A035336) or 1, 4, 6, 9, 12, ... (A003622), we get the Wythoff array extended by 1 column (A287869) or 2 columns (A287870) respectively.

Similarly, using A035338 truncates by 1 column; and in general if S_k is column k of the Wythoff array then the rows here numbered by S_k form an array A_k that starts with column k-2 of the Wythoff array. (A_0 and A_1 are the 2 extended arrays mentioned above.) As every positive integer occurs exactly once in the Wythoff array, every row except row 0 of A(.,.) is a row of exactly one such A_k.

Columns 4 onwards match certain columns of the multiplication table for Knuth's Fibonacci (or circle) product (extended variant - see A135090 and formula below).

For k > 0, the first row to contain k is A348853(k).