cp's OEIS Frontend

T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.

From Gus Wiseman, Jul 10 2025: (Start)

Also the number of integer partitions of n with k maximal subsequences of consecutive parts not decreasing by 1 (anti-runs). For example, row n = 8 counts partitions with the following anti-runs:

((8)) ((3,3),(2)) ((3),(2,2),(1))

((4,4)) ((4),(3,1)) ((3),(2),(1,1,1))

((5,3)) ((5,2),(1))

((6,2)) ((4,2),(1,1))

((7,1)) ((2,2,2),(1,1))

((4,2,2)) ((2,2),(1,1,1,1))

((6,1,1)) ((2),(1,1,1,1,1,1))

((2,2,2,2))

((3,3,1,1))

((5,1,1,1))

((4,1,1,1,1))

((3,1,1,1,1,1))

((1,1,1,1,1,1,1,1))

(End)

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003

Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003

Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005

a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006

Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010

Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).

Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

Number of terms of A164710 with exactly n+1 binary digits. - Robert Israel, Nov 09 2015

From Gus Wiseman, Jun 23 2025: (Start)

This is the number of subsets of {1..n} with all equal lengths of runs of consecutive elements increasing by 1. For example, the runs of S = {1,2,5,6,8,9} are ((1,2),(5,6),(8,9)), with lengths (2,2,2), so S is counted under a(9). The a(0) = 1 through a(4) = 14 subsets are:

{} {} {} {} {}

{1} {1} {1} {1}

{2} {2} {2}

{1,2} {3} {3}

{1,2} {4}

{1,3} {1,2}

{2,3} {1,3}

{1,2,3} {1,4}

{2,3}

{2,4}

{3,4}

{1,2,3}

{2,3,4}

{1,2,3,4}

(End)

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006

Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012

Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012

A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x). See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012

The matrix inverse starts

1;

-1, 1;

-1, -1, 1;

1, -2, -1, 1;

3, 1, -3, -1, 1;

1, 6, 1, -4, -1, 1;

-7, 4, 10, 1, -5, -1, 1;

-13, -13, 8, 15, 1, -6, -1, 1;

3, -31, -23, 13, 21, 1, -7, -1, 1; - R. J. Mathar, Mar 15 2013

Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025