cp's OEIS Frontend

As a number triangle, this is given by T(n,k)=sum{j=0..n, C(n,j)(-1)^(n-j)sum{i=0..j, C(j+k,i-k)}}. - Paul Barry, Aug 23 2004

As a number triangle, this is the Riordan array (1/(1-x), x(1+x)) with T(n,k)=sum{i=0..n, binomial(k,i-k)}. Diagonal sums are then A023434(n+1). - Paul Barry, Feb 16 2005

Form partial sums across rows of square array of binomial coefficients A026729; see also A008949. - Philippe Deléham, Aug 28 2005

Square array A026729 -> Partial sums across rows

1 0 0 0 0 0 0 . . . . 1 1 1 1 1 1 1 . . . . . .

1 1 0 0 0 0 0 . . . . 1 2 2 2 2 2 2 . . . . . .

1 2 1 0 0 0 0 . . . . 1 3 4 4 4 4 4 . . . . . .

1 3 3 1 0 0 0 . . . . 1 4 7 8 8 8 8 . . . . . .

For other Whitney numbers see A007799.

W(n,k) is the number of length k binary sequences containing no more than n 1's. - Geoffrey Critzer, Mar 15 2010

From Emeric Deutsch, Jun 15 2010: (Start)

Viewed as a number triangle, T(n,k) is the number of internal nodes of the Fibonacci tree of order n+2 at level k. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.

(End)

Named after the American mathematician Hassler Whitney (1907-1989). - Amiram Eldar, Jun 13 2021

Starting with "1" = eigensequence of a triangle with bin(n,4), A000332 as the left border: (1, 5, 15, 35, 70, ...) and the rest 1's. - Gary W. Adamson, Jul 24 2010

The Kn25 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the four leading zeros. - Johannes W. Meijer, Aug 14 2011

(1 + 6x + 22x^2 + 64x^3 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x + 7x^2 + 15x^3 + ...). - Gary W. Adamson, Mar 14 2012

The sequence starting (1, 6, 22, ...) is the binomial transform of A171418 and starting (0, 0, 0, 1, 6, 22, ...) is the binomial transform of (0, 0, 0, 1, 2, 2, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015

Number of binary sequences with at least four 0's. - Enrique Navarrete, Jul 23 2025

Number of words of length n+1 where first element is from {0,1,2}, other elements are from {0,1} and sequence does not decrease (for n=2 there are 3*2^2 sequences, but 000, 100, 110, 111, 200, 210, 211 decrease, so a(2) = 12-7 = 5).

Number of subgroups of C_(2^n) X C_(2^n) (see A060724).

Starting with 1 = row sums of triangle A054582. - Gary W. Adamson, Jun 23 2008

Starting with "1" equals the eigensequence of a triangle with integer squares (1, 4, 9, 16, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

(1 + 2x + 2x^2 + 2x^3 + ...)*(1 + 3x + 7x^2 + 15x^3 + ...) = (1 + 5x + 15x^2 + 37x^3 + ...). - Gary W. Adamson, Mar 14 2012

The partial sums of A033484. - J. M. Bergot, Oct 03 2012

Binomial transform is 0, 1, 7, 33, ... (shifted A066810); inverse binomial transform is 0, 1, 3, 3, ... (3 repeated). - R. J. Mathar, Oct 05 2012

Define a triangle by T(n,0) = n*(n+1) + 1, T(n,n) = n + 1, and T(r,c) = T(r-1,c-1) + T(r-1,c) otherwise; then a(n+1) is the sum of the terms of row n. - J. M. Bergot, Mar 30 2013

Starting with "1" are also the antidiagonal sums of the array formed by partial sums of integer squares (1, 4, 9, 16, ...). - Luciano Ancora, Apr 24 2015

Sums of 2 adjacent terms in diagonal k=2 of Eulerian triangle A008292. I.e., T(n,2)+T(n-1,2) for n > 0. Also, 4th NW-SE diagonal of A126277. In other words, a(n) = A000295(n) + A000295(n+1). - Gregory Gerard Wojnar, Sep 30 2018

a(0)=1, otherwise row sums of A179748.

For n>=1 cumulative sums of A008930.

a(n) is proportional to A048651*A000079. The error (a(n)-A048651*A000079) divided by sequence A186425 tends to the golden ratio A001622. This can be seen when using about 1000 decimals of the constant A048651 = 0.2887880950866024212... - [Mats Granvik, Jan 01 2015]

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:

(1) (2) (3) (4) (5)

(11) (21) (31) (41)

(111) (112) (212)

(211) (311)

(1111) (1112)

(1121)

(2111)

(11111)

(End)

Eigensequence of the triangle = A001861. - Gary W. Adamson, Apr 17 2009

Ways of placing n+1 labeled balls into 2 indistinguishable boxes with at least 2 balls in each box.

2^a(n) is an integer of the form 1/(2 - Sum_{i=1..m} i/2^i). - Benoit Cloitre, Oct 25 2002

Number of permutations avoiding 13-2 that contain the pattern 23-1 exactly twice.

Cost of ternary maximum height Huffman tree with N internal nodes (non-leaves) for minimizing absolutely ordered sequences of size n = 2N + 1. - Alex Vinokur (alexvn(AT)barak-online.net), Nov 02 2004

a(n) is the number of Dyck n-paths whose third upstep initiates the last long ascent, n >= 1. A long ascent is one consisting of 2 or more upsteps. For example, a(3)=3 counts UUDuUDDD, UDUDuUDD, UUDDuUDD (third upstep in small type). - David Callan, Dec 08 2004

Subsequence of A158581; A000120(a(n)) > 1. - Reinhard Zumkeller, Apr 16 2009

Number of vertices of the tropical Grassmannian simplicial complex G(2,n), related to phylogenetic trees. - Tom Copeland, Oct 03 2011

(Conjecture) Let a(2)=0. For n > 2, let N = 2*n + 1. For each n, define the n X n tridiagonal unit-primitive matrix (see [Jeffery]) A_{N,1}=[0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1] associated with N. Define the n-dimensional column vectors V_N = [v_1,v_2,...,v_n]^T = [A_{N,1}]^n*[1,1,...,1]^T, where [.]^T denotes matrix transpose and [1,...,1] is the n-dimensional unit vector. Let (v_k)N denote the k-th element of V_N, k in {1,...,n}. Then a(n) = (v(n-2))N. - _L. Edson Jeffery, Jan 20 2012

For n>0, (a(n)) is row 3 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

For n>2, a(n-2) is the number of connected induced (non-null) subgraphs of the n-centipede graph. - Giovanni Resta, May 04 2017

a(n) is the number of maximal boundary strata of the moduli space of stable rational curves with n+1 marked points. The closures of the maximal boundary strata are called the irreducible boundary divisors of the moduli space; see Cavalieri Section 2.1. - Harry Richman, Aug 13 2024

A000295 (Eulerian numbers) gives the row sums.

Write A004736 and Pascal's triangle as infinite lower triangular matrices A and B; then A*B is this triangle.

From Peter Luschny, Apr 10 2011: (Start)

A slight variation has a combinatorial interpretation: remove the last column and the second one from Pascal's triangle. Let P(m, k) denote the set partitions of {1,2,..,n} with the following properties:

(a) Each partition has at least one singleton block;

(c) k is the size of the largest block of the partition;

(b) m = n - k + 1 is the number of parts of the partition.

Then A000295(n) = Sum_{k=1..n} card(P(n-k+1,k)).

For instance, A000295(4) = P(4,1) + P(3,2) + P(2,3) + P(1,4) = card({1|2|3|4}) + card({1|2|34, 1|3|24,1|4|23, 2|3|14, 2|4|13, 3|4|12}) + card({1|234, 2|134, 3|124, 4|123}) = 1 + 6 + 4 = 11.

This interpretation can be superimposed on the sequence by changing the offset to 1 and adding the value 1 in front. The triangle then starts

1;

1, 3;

1, 6, 4;

1, 10, 10, 5;

1, 15, 20, 15, 6;

...

(End)

Diagonal sums are A001924(n+1). - Philippe Deléham, Jan 11 2014

Relation to K-theory: T acting on the column vector (d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168, A111492, A238363, and A135278. - Tom Copeland, Apr 11 2014

From Gary W. Adamson, Jul 24 2010: (Start)

Starting with "1" = eigensequence of a triangle with binomial C(n,5):

(1, 6, 21, 56, ...) as the left border and the rest 1's. (End)

The Kn26 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the five leading zeros. - Johannes W. Meijer, Aug 15 2011

Starting (0, 0, 0, 0, 1, 7, 29, ...), this is the binomial transform of (0, 0, 0, 0, 1, 2, 2, 2, ...). Starting (1, 7, 29, ...), this is the binomial transform of (1, 6, 16, 26, 31, 32, 32, 32, ...). - Gary W. Adamson, Jul 28 2015

Starting with "1", equals the eigensequence of a triangle with A000579 = binomial(n,6) = (1, 7, 28, 84, 210, ...) as the left column and the rest 1's. - Gary W. Adamson, Jul 24 2010

p divides a(p+1) for all prime p except 3. p^2 divides a(p+1) for prime p in A123374.

2 divides a(n) for n = {2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, ...}.

2^2 divides a(n) for n = {2, 3, 6, 7, 8, 10, 11, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 30, 31, 32, 34, 35, 38, 39, 40, 42, 43, 46, 47, 48, 50, ...}.

2^3 divides a(n) for n = {2, 3, 6, 7, 10, 11, 14, 15, 16, 18, 19, 22, 23, 26, 27, 30, 31, 32, 34, 35, 38, 39, 42, 43, 46, 47, 48, 50, ...}.

2^4 divides a(n) for n = {7, 14, 15, 18, 23, 30, 31, 32, 34, 39, 46, 47, 50, ...}.

2^5 divides a(n) for n = {15, 30, 31, 34, 47, 62, 63, 64, 66, 79, 94, 95, 98, ...}.

2^6 divides a(n) for n = {31, 62, 63, 66, 95, ...}.

2^7 divides a(n) for n = {63, 126, 127, 130, ...}.

It appears that for k > 2 the least few n such that a(n) is divisible by 2^(k+1) are n = {(2^k-1), 2*(2^k-1), 2*(2^k-1)+1, 2*(2^k-1)+3, 3*(2^k-1)+2, 4*(2^k-1)+2, 4*(2^k-1)+3, 4*(2^k-1)+4, 4*(2^k-1)+6, 5*(2^k-1)+4, 6*(2^k-1)+4, 6*(2^k-1)+5, 6*(2^k-1)+8, ...}. - Alexander Adamchuk, Oct 08 2006

Numbers n that divide a(n) are listed in A014741. - Alexander Adamchuk, Nov 03 2006