0,2

Row sums of A004248, A009998, A009999.

First differences are in A047970.

First differences of A103439.

Antidiagonal sums of array A003992.

a(n-1), for n>=1, is the number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, that are simultaneously projections as maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x); see example and the two comments (Arndt, Apr 30 2011 Jan 04 2013) in A000110. - Joerg Arndt, Mar 07 2015

Number of finite sequences s of length n+1 whose discriminator sequence is s itself. Here the discriminator sequence of s is the one where the n-th term (n>=1) is the least positive integer k such that the first n terms are pairwise incongruent, modulo k. - Jeffrey Shallit, May 17 2016

From Gus Wiseman, Jan 08 2019: (Start)

Also the number of set partitions of {1,...,n+1} whose minima form an initial interval of positive integers. For example, the a(3) = 9 set partitions are:

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

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

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

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

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

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

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

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

{{1,2,3,4}}

Missing from this list are:

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

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

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

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

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

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

(End)

a(n) is the number of m-tuples of nonnegative integers less than or equal to n-m (including the "0-tuple"). - Mathew Englander, Apr 11 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.

Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 12.

Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.

Sajed Haque, Discriminators of Integer Sequences, 2017, See p. 33 Corollary 29.

Mathematics Stack Exchange, Asymptotics of ..., 2011.

Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019. See p. 3.

Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See p. 4.

a(n) = A003101(n) + 1.

G.f.: Sum_{n>=0} x^n/(1 - (n+1)*x). - Paul D. Hanna, Sep 13 2011

G.f.: G(0) where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

E.g.f.: Sum_{n>=0} Integral^n exp((n+1)*x) dx^n, where Integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013

O.g.f.: Sum_{n>=0} n! * x^n/(1-x)^(n+1) / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Jul 20 2014

a(n) = A101494(n+1,0). - Vladimir Kruchinin, Apr 01 2015

a(n-1) = Sum_{k = 1..n} k^(n-k). - Gus Wiseman, Jan 08 2019

log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021

a(n) ~ sqrt(2*Pi/(n+1 + (n+1)/w(n))) * ((n+1)/w(n))^(n+2 - (n+1)/w(n)), where w(n) = LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 23*x^4 + 66*x^5 + 210*x^6 + ...

where we have the identity:

A(x) = 1/(1-x) + x/(1-2*x) + x^2/(1-3*x) + x^3/(1-4*x) + x^4/(1-5*x) + ...

is equal to

A(x) = 1/(1-x) + x/((1-x)^2*(1+x)) + 2!*x^2/((1-x)^3*(1+x)*(1+2*x)) + 3!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^4/((1-x)^5*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

From Joerg Arndt, Mar 07 2015: (Start)

The a(5-1) = 23 RGS described in the comment are (dots denote zeros):

01: [ . . . . . ]

02: [ . 1 . . . ]

03: [ . 1 . . 1 ]

04: [ . 1 . 1 . ]

05: [ . 1 . 1 1 ]

06: [ . 1 1 . . ]

07: [ . 1 1 . 1 ]

08: [ . 1 1 1 . ]

09: [ . 1 1 1 1 ]

10: [ . 1 2 . . ]

11: [ . 1 2 . 1 ]

12: [ . 1 2 . 2 ]

13: [ . 1 2 1 . ]

14: [ . 1 2 1 1 ]

15: [ . 1 2 1 2 ]

16: [ . 1 2 2 . ]

17: [ . 1 2 2 1 ]

18: [ . 1 2 2 2 ]

19: [ . 1 2 3 . ]

20: [ . 1 2 3 1 ]

21: [ . 1 2 3 2 ]

22: [ . 1 2 3 3 ]

23: [ . 1 2 3 4 ]

(End)

a:= n-> add((n+1-j)^j, j=0..n): seq(a(n), n=0..23); # Zerinvary Lajos, Apr 18 2009

Table[Sum[(n-k+1)^k, {k, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Apr 01 2015 *)

(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/(1-(m+1)*x+x*O(x^n))), n)} /* Paul D. Hanna, Sep 13 2011 */

(PARI) {INTEGRATE(n, F)=local(G=F); for(i=1, n, G=intformal(G)); G}

{a(n)=local(A=1+x); A=sum(k=0, n, INTEGRATE(k, exp((k+1)*x+x*O(x^n)))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Dec 28 2013

for(n=0, 30, print1(a(n), ", "))

(PARI)

{a(n)=polcoeff( sum(m=0, n, m!*x^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x +x*O(x^n))), n)} /* From o.g.f. (Paul D. Hanna, Jul 20 2014) */

for(n=0, 25, print1(a(n), ", "))

(Haskell)

a026898 n = sum $ zipWith (^) [n + 1, n .. 1] [0 ..]

-- Reinhard Zumkeller, Sep 14 2014

(Magma) [(&+[(n-k+1)^k: k in [0..n]]): n in [0..50]]; // Stefano Spezia, Jan 09 2019

(SageMath) [sum((n-j+1)^j for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

Cf. A000110, A000258, A000670, A003101, A008277, A038125, A062810.

Cf. A105795, A179928, A287215, A287216.

Sequence in context: A261134 A117419 A124461 * A088930 A225588 A089844

Adjacent sequences: A026895 A026896 A026897 * A026899 A026900 A026901

a(23)-a(25) from Paul D. Hanna, Dec 28 2013

approved