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A026898 a(n) = Sum_{k=0..n} (n-k+1)^k. 20
1, 2, 4, 9, 23, 66, 210, 733, 2781, 11378, 49864, 232769, 1151915, 6018786, 33087206, 190780213, 1150653921, 7241710930, 47454745804, 323154696185, 2282779990495, 16700904488706, 126356632390298, 987303454928973, 7957133905608837, 66071772829247410 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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)

LINKS

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

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

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

FORMULA

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

EXAMPLE

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)

MAPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A003101, A038125, A062810, A287216.

Cf. A000110, A000258, A000670, A008277, A105795, A287215.

Sequence in context: A261134 A117419 A124461 * A088930 A225588 A089844

Adjacent sequences:  A026895 A026896 A026897 * A026899 A026900 A026901

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified August 13 00:27 EDT 2020. Contains 336441 sequences. (Running on oeis4.)