cp's OEIS Frontend

Length of n-th row given by A000120(n);

Min of n-th row given by A001511(n);

Sum of n-th row given by A029931(n);

Product of n-th row given by A096111(n);

Max of n-th row given by A113473(n);

Numerator of sum of reciprocals of n-th row given by A116416(n);

Denominator of sum of reciprocals of n-th row given by A116417(n);

LCM of n-th row given by A271410(n).

The first appearance of n is at A001787(n - 1).

n-th row begins at index A000788(n - 1) for n > 0.

Also the reversed positions of 1's in the reversed binary expansion of n. Also the reversed partial sums of the n-th composition in standard order (row n of A066099). Reversing rows gives A048793. - Gus Wiseman, Jan 17 2023

Right side of the binomial sum Sum_{i = 0..n} (n-2*i)^2 * binomial(n, i) = n*2^n. - Yong Kong (ykong(AT)curagen.com), Dec 28 2000

Let W be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x, y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y, or y is a proper subset of x and there are no z in P(A) such that y is a proper subset of z and z is a proper subset of x. Then a(n) = |W|. - Ross La Haye, Sep 26 2007

Partial sums give A036799. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

a(n) = n with the bits shifted to the left by n places (new bits on the right hand side are zeros). - Indranil Ghosh, Jan 05 2017

Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017

Also the circumference of the n-cube connected cycle graph. - Eric W. Weisstein, Sep 03 2017

a(n) is also the number of derangements in S_{n+3} with a descent set of {i, i+1} such that i ranges from 1 to n-2. - Isabella Huang, Mar 17 2018

a(n-1) is also the number of multiplications required to compute the permanent of general n X n matrices using Glynn's formula (see Theorem 2.1 in Glynn). - Stefano Spezia, Oct 27 2021

This infinite matrix is the square of the Pascal matrix (A007318) whose rows are [ 1,0,... ], [ 1,1,0,... ], [ 1,2,1,0,... ], ...

As an upper right triangle, table rows give number of points, edges, faces, cubes,

4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009

Number of different partial sums of 1+[1,1,2]+[2,2,3]+[3,3,4]+[4,4,5]+... with entries that are zero removed. - Jon Perry, Jan 01 2004

Row sums are powers of 3 (A000244), antidiagonal sums are Pell numbers (A000129). - Gerald McGarvey, May 17 2005

Riordan array (1/(1-2x), x/(1-2x)). - Paul Barry, Jul 28 2005

T(n,k) is the number of elements of the Coxeter group B_n with descent set contained in {s_k}, 0<=k<=n-1. For T(n,n), we interpret this as the number of elements of B_n with empty descent set (since s_n does not exist). - Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006

Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then T(n,k) = the number of elements (x,y) of S for which y has exactly k more elements than x. - Ross La Haye, Oct 12 2007

T(n,k) is number of paths in the first quadrant going from (0,0) to (n,k) using only steps B=(1,0) colored blue, R=(1,0) colored red and U=(1,1). Example: T(3,2)=6 because we have BUU, RUU, UBU, URU, UUB and UUR. - Emeric Deutsch, Nov 04 2007

T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (0,1), and two kinds of step (1,0). - Joerg Arndt, Jul 01 2011

T(i,j) is the number of i-permutations of {1,2,3} containing j 1's. Example: T(2,1)=4 because we have 12, 13, 21 and 31; T(3,2)=6 because we have 112, 113, 121, 131, 211 and 311. - Zerinvary Lajos, Dec 21 2007

Triangle of coefficients in expansion of (2+x)^n. - N-E. Fahssi, Apr 13 2008

Sum of diagonals are Jacobsthal-numbers: A001045. - Mark Dols, Aug 31 2009

Triangle T(n,k), read by rows, given by [2,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

Eigensequence of the triangle = A004211: (1, 3, 11, 49, 257, 1539, ...). - Gary W. Adamson, Feb 07 2010

f-vectors ("face"-vectors) for n-dimensional cubes [see e.g., Hoare]. (This is a restatement of Bottomley's above.) - Tom Copeland, Oct 19 2012

With P = Pascal matrix, the sequence of matrices I, A007318, A038207, A027465, A038231, A038243, A038255, A027466 ... = P^0, P^1, P^2, ... are related by Copeland's formula below to the evolution at integral time steps n= 0, 1, 2, ... of an exponential distribution exp(-x*z) governed by the Fokker-Planck equation as given in the Dattoli et al. ref. below. - Tom Copeland, Oct 26 2012

The matrix elements of the inverse are T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 12 2013

Unsigned diagonals of A133156 are rows of this array. - Tom Copeland, Oct 11 2014

Omitting the first row, this is the production matrix for A039683, where an equivalent differential operator can be found. - Tom Copeland, Oct 11 2016

T(n,k) is the number of functions f:[n]->[3] with exactly k elements mapped to 3. Note that there are C(n,k) ways to choose the k elements mapped to 3, and there are 2^(n-k) ways to map the other (n-k) elements to {1,2}. Hence, by summing T(n,k) as k runs from 0 to n, we obtain 3^n = Sum_{k=0..n} T(n,k). - Dennis P. Walsh, Sep 26 2017

Since this array is the square of the Pascal lower triangular matrix, the row polynomials of this array are obtained as the umbral composition of the row polynomials P_n(x) of the Pascal matrix with themselves. E.g., P_3(P.(x)) = 1 P_3(x) + 3 P_2(x) + 3 P_1(x) + 1 = (x^3 + 3 x^2 + 3 x + 1) + 3 (x^2 + 2 x + 1) + 3 (x + 1) + 1 = x^3 + 6 x^2 + 12 x + 8. - Tom Copeland, Nov 12 2018

T(n,k) is the number of 2-compositions of n+1 with some zeros allowed that have k zeros; see the Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020

Also the convolution triangle of A000079. - Peter Luschny, Oct 09 2022

The sum of the terms of the inverse of the binomial coefficients, 1/C(k,n), with alternating signs, equals an irrational number which is expressed as m * log(2) - r, where m is the integer n*2^(n-1) = A001787(n), n>=1, and r is rational. a(n) = r * (n-1)!.

Let M_n be the n X n matrix m_(i,j) = 2 + abs(i-j) then det(M_n) = (-1)^(n-1)*a(n-1). - Benoit Cloitre, May 28 2002

a(n) is the number of triangulations of a regular (n+3)-gon in which every triangle shares at least one side with the polygon itself. - David Callan, Mar 25 2004

Number of compositions of j+n, j>n and j the maximum part. E.g. a(4) is derived from the number of compositions of, for example: 54(2), 531(6), 522(3), 5211(12) and 51111(5) giving 2+6+3+12+5=28. - Jon Perry, Sep 13 2005

If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n+1) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

Generated from iterates of M * [1,1,1,...], where M = an infinite triadiagonal matrix with (1,1,1,...) in the main and superdiagonals and (1,0,0,0,...) in the subdiagonal. - Gary W. Adamson, Jan 04 2009

a(n) is the number of weak compositions of n with exactly 1 part equal to 0. - Milan Janjic, Jun 27 2010

An elephant sequence, see A175654. For the corner squares 16 A[5] vectors, with decimal values between 19 and 400, lead to this sequence. For the central square these vectors lead to the companion sequence A045891 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010

Equals first finite difference row of A001792: (1, 3, 8, 20, 48, 112, ...). - Gary W. Adamson, Oct 26 2010

With alternating signs the g.f. is: (1 + x)^2/(1 + 2*x)^2.

Number of 132-avoiding permutations of [n+2] containing exactly one 213 pattern. - David Scambler, Nov 07 2011

a(n) is the number of 1's in all compositions of n+1 = the number of 2's in all compositions of n+2 = the number of 3's in all compositions of n+3 = ... So the partial sums = A001792. - Geoffrey Critzer, Feb 12 2012

Also number of compositions of n into 2 sorts of parts where all parts of the first sort precede all parts of the second sort; see example. - Joerg Arndt, Apr 28 2013

a(n) is also the difference of the total number of parts between all compositions of n+1 and all compositions of n. The equivalent sequence for partitions is A138137. - Omar E. Pol, Aug 28 2013

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^2; see A291000. - Clark Kimberling, Aug 24 2017

For a composition of n, the total number of runs of parts of size k is a(n-k) - a(n-2k). - Gregory L. Simay, Feb 17 2018

a(n) is the number of binary trees on n+1 nodes that are isomorphic to a path graph. The ratio of a(n)/A000108(n+1) gives the probability that a random Catalan tree on n+1 nodes is isomorphic to a path graph. - Marcel K. Goh, May 09 2020

a(n) is the number of words of length n over the alphabet {0,1,2} such that the first letter is not 2 and the last 1 occurs before the first 0. - Henri Mühle, Mar 08 2021

Also the number of "special permutations" in the Weng and Zagier reference. - F. Chapoton, Sep 30 2022

a(n-k) is the total number of runs of 1s of length k over all binary n-strings. - Félix Balado, Dec 11 2022

Partial sums of A000120.

The graph of this sequence is a version of the Takagi curve: see Lagarias (2012), Section 9, especially Theorem 9.1. - N. J. A. Sloane, Mar 12 2016

a(n-1) is the largest possible number of ordered pairs (a,b) such that a/b is a prime in a subset of the positive integers with n elements. - Yifan Xie, Feb 21 2025

Figurate numbers based on 7-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 15 of these numbers. - Jonathan Vos Post, Nov 28 2004

a(n) is the number of terms in the expansion of (Sum_{i=1..8} a_i)^n. - Sergio Falcon, Feb 12 2007

Product of seven consecutive numbers divided by 7!. - Artur Jasinski, Dec 02 2007

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 6 elements, which is 3*C(n+1,7) (for n>=6), hence a(n) = 3*C(n+1,7) = 3*A000580(n+1). - Serhat Bulut, Mar 13 2015

Partial sums of A000579. In general, the iterated sums S(m, n) = Sum_{j=1..n} S(m-1, j) with input S(1, n) = A000217(n) = 1 + 2 + ... + n are S(m, n) = risefac(n, m+1)/(m+1)! = binomial(n+m, m+1) = Sum_{k = 1..n} risefac(k, m)/m!, with the rising factorials risefac(x, m):= Product_{j=0..m-1} (x+j), for m >= 1. Such iterated sums of arithmetic progressions have been considered by Narayana Pandit (see The MacTutor History of Mathematics archive link, and the Gottwald et al. reference, p. 338, where the name Narayana Daivajna is also used). - Wolfdieter Lang, Mar 20 2015

a(n) = fallfac(n,7)/7! = binomial(n, 7) is also the number of independent components of an antisymmetric tensor of rank 7 and dimension n >= 7 (for n=1..6 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015

From Juergen Will, Jan 02 2016: (Start)

Number of compositions (ordered partitions) of n+1 into exactly 8 parts.

Number of weak compositions (ordered weak partitions) of n-7 into exactly 8 parts. (End)

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003

Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010

Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015

Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015

Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021

Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

a(n) also gives number of 0's in binary numbers 1 to 111..1 (n+1 bits). - Stephen G Penrice, Oct 01 2000

Numerator of m(n) = (m(n-1)+n)/2, m(0)=0. Denominator is A000079. - Reinhard Zumkeller, Feb 23 2002

a(n) is the number of directed column-convex polyominoes of area n+2 having along the lower contour exactly one vertical step that is followed by a horizontal step (a reentrant corner). - Emeric Deutsch, May 21 2003

a(n) is the number of bits in binary numbers from 1 to 111...1 (n bits). Partial sums of A001787. - Emeric Deutsch, May 24 2003

Genus of graph of n-cube = a(n-3) = 1+(n-4)*2^(n-3), n>1.

Sum of ordered partitions of n where each element is summed via T(e-1). See A066185 for more information. - Jon Perry, Dec 12 2003

a(n-2) is the number of Dyck n-paths with exactly one peak at height >= 3. For example, there are 5 such paths with n=4: UUUUDDDD, UUDUUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD. - David Callan, Mar 23 2004

Permutations in S_{n+2} avoiding 12-3 that contain the pattern 13-2 exactly once.

a(n) is prime for n = 2, 3, 7, 27, 51, 55, 81. a(n) is semiprime for n = 4, 5, 6, 8, 9, 10, 11, 13, 15, 19, 28, 32, 39, 57, 63, 66, 75, 97. - Jonathan Vos Post, Jul 18 2005

A member of the family of sequences defined by a(n) = Sum_{i=1..n} i*[c(1)*...*c(r)]^(i-1). This sequence has c(1)=2, A014915 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008

Starting with 1 = row sums of A023758 as a triangle by rows: [1; 2,3; 4,6,7; 8,12,14,15; ...]. - Gary W. Adamson, Jul 18 2008

Equivalent formula given in Brehm: for each q >= 3 there exists a polyhedral map M_q of type {4, q} with [number of vertices] f_0 = 2^q and [genus] g = (2^(q-3))*(q-4) + 1 such that M_q and its dual have polyhedral embeddings in R^3 [McMullen et al.]. - Jonathan Vos Post, Jul 25 2009

Sums of rows of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010

This sequence is related to A000079 by a(n) = n*A000079(n)-Sum_{i=0..n-1} A000079(i). - Bruno Berselli, Mar 06 2012

(1 + 5*x + 17*x^2 + 49*x^3 + ...) = (1 + 2*x + 4*x^2 + 8*x^3 + ...) * (1 + 3*x + 7*x^2 + 15*x^3 + ...). - Gary W. Adamson, Mar 14 2012

The first barycentric coordinate of the centroid of Pascal triangles, assuming that numbers are weights, is A000295(n+1)/A000337(n), no matter what the triangle sides are. See attached figure. - César Eliud Lozada, Nov 14 2014

a(n) is the n-th number that is a sum of n positive n-th powers for n >= 1. a(4) = 49 = A003338(4). - Alois P. Heinz, Aug 01 2020

a(n) is the sum of the largest elements of all subsets of {1,2,..,n}. For example, a(3)=17; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of the largest elements is 17. - Enrique Navarrete, Aug 20 2020

a(n-1) is the sum of the second largest elements of the subsets of {1,2,..,n} that contain n. For example, for n = 4, a(3)=17; the subsets of {1,2,3,4} that contain 4 are {4}, {1,4}, {2,4}, {3,4}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, and the sum of the second largest elements is 17. - Enrique Navarrete, Aug 24 2020

a(n-1) is also the sum of diameters of all subsets of {1,2,...,n} that contain n. For example, for n = 4, a(3)=17; the subsets of {1,2,3,4} that contain 4 are {4}, {1,4}, {2,4}, {3,4}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}; the diameters of these sets are 0,3,2,1,3,3,2,3 and the sum is 17. - Enrique Navarrete, Sep 07 2020

a(n-1) is also the number of additions required to compute the permanent of general n X n matrices using trellis methods (see Theorems 5 and 6, pp. 10-11 in Kiah et al.). - Stefano Spezia, Nov 02 2021