cp's OEIS Frontend

Binomial transform of 1+n*(n+1)/2, A000124.

Number of 123-avoiding ternary words of length n-1.

Row sums of triangle A134247. Also double binomial transform of (1, 1, 1, 0, 0, 0, ...). - Gary W. Adamson, Oct 15 2007

Equals row sums of triangle A144333. - Gary W. Adamson, Sep 18 2008

Rows, of the square array, are three-fold convolutions of sequences of powers.

Binomial transform of A000124 (when this begins 1,1,2,4,7,...).

2nd binomial transform of (1,0,1,0,0,0,...).

Case k=2 where a(n,k) = k^n(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.

Previous name was: Triangle read by rows: T(n,k)=derivative of the q-binomial coefficient [n,k] evaluated at q=1 (0<=k<=n). - N. J. A. Sloane, Jan 06 2016

For example, T(5,2)=30 because [5,2] = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1 with derivative 6q^5 + 5q^4 + 8q^3 + 6q^2 + 4q + 1, having value 30 at q=1. - Emeric Deutsch, Apr 22 2007

Sum of entries in n-th antidiagonal = n(n-1)2^(n-3) = A001788(n-1).

T(n,k) = A094305(n-2, k-1) for n >= 2, k >= 1.

T(n,k) is total number of pips on a set of generalized linear dominoes with n cells (rather than two) and with the number of pips in each cell running from 0 to k (rather than 6). T(2,6) = 168 gives the total number of pips on a standard set of dominoes. We regard a generalized linear domino with n cells and up to k pips per cell as an ordered n-tuple [i_1, i_2, ..., i_n] with 0 <= i_1 <= i_2 <= ... <= i_n <= k. - Alan Shore and N. J. A. Sloane, Jan 06 2016

T(n,k) can also be written more symmetrically as the trinomial coefficient (n+k; n-1, k-1, 2). - N. J. A. Sloane, Jan 06 2016

As a triangle read by rows, T(n,k) is the total number of inversions over all length n binary words having exactly k 1's. T(n,k) is also the total area above all North East lattice paths from the origin to the point (k,n-k). - Geoffrey Critzer, Mar 22 2018

With a different offset, number of n-permutations of 9 objects:

p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's. Example: a(1)=72 because we have

uuuuuuuup, uuuuuuupu, uuuuuupuu, uuuuupuuu, uuuupuuuu, uuupuuuuu, uupuuuuuu, upuuuuuuu, puuuuuuuu,

uuuuuuuur, uuuuuuuru, uuuuuuruu, uuuuuruuu, uuuuruuuu, uuuruuuuu, uuruuuuuu, uruuuuuuu, ruuuuuuuu,

uuuuuuuus, uuuuuuusu, uuuuuusuu, uuuuusuuu, uuuusuuuu, uuusuuuuu, uusuuuuuu, usuuuuuuu, suuuuuuuu,

uuuuuuuut, uuuuuuutu, uuuuuutuu, uuuuutuuu, uuuutuuuu, uuutuuuuu, uutuuuuuu, utuuuuuuu, tuuuuuuuu,

uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu,

uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu, zuuuuuuuu,

uuuuuuuux, uuuuuuuxu, uuuuuuxuu, uuuuuxuuu, uuuuxuuuu, uuuxuuuuu, uuxuuuuuu, uxuuuuuuu, xuuuuuuuu,

uuuuuuuuy, uuuuuuuyu, uuuuuuyuu, uuuuuyuuu, uuuuyuuuu, uuuyuuuuu, uuyuuuuuu, uyuuuuuuu, yuuuuuuuu.

The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - Peter Luschny, Mar 13 2009

T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.

From Peter Bala, Jul 22 2014: (Start)

Exponential Riordan array [(1+x)*exp(x), x*exp(x)].

Let M = A093375, the exponential Riordan array [(1+x)*exp(x), x], and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0 M/

having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... - see the Example section. (End)

The Bell transform of n+1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

The first element of each partition is given weight 0.

Consider the partitions of n, e.g., n=5. For each partition sum T(e-1) and sum all these. E.g., 5 -> T(4)=10, 41 -> T(3)+T(0)=6, 32 -> T(2)+T(1)=4, 311 -> T(2)+T(0)+T(0)=3, 221 -> T(1)+T(1)+T(0)=2, 21111 ->1 and 11111 ->0. Summing, 10+6+4+3+2+1+0 = 26 as desired. - Jon Perry, Dec 12 2003

Also equals the sum of f(p) over the partitions p of n, where f(p) is obtained by replacing each part p_i of partition p by p_i*(p_i-1)/2. See I. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition, p. 3, eqn (1.5) and (1.6). - Wouter Meeussen, Sep 25 2014

From Enrique Navarrete, Jun 13 2023: (Start)

a(n) is the number of ways to partition the set [n]={1,2,...,n} into two sets S,T and select 2 elements in total (from either S or T or both).

Example. For n=4, sample partitions are given (where S(i),T(j) means i elements are selected from S, j elements are selected from T):

S={ }, T={1,2,3,4}: partition [4] in 1 way, S(0),T(2) (6 ways);

S={1}, T={2,3,4}: partition [4] in 4 such ways, S(1),T(1) or S(0),T(2) (24 ways);

S={1,2}, T={3,4}: partition [4], in such 6 ways, S(1),T(1) or S(0),T(2) or S(2),T(0) (36 ways);

S={1,2,3}, T={4}: partition [4] in 4 such ways, S(1),T(1) or S(2),T(0) (24 ways);

S={1,2,3,4}, T={ }: partition [4] in 1 way, S(2),T(0) (6 ways). (End)

Number of inversions in all ternary words of length n on {0,1,2}. Example: a(2)=3 because each of the words 10,20,21 has one inversion and the words 00,01,02,11,12,22 have no inversions. a(n)=3*A027472(n+1). a(n)=Sum(k*A129529(n,k),k>=0).