A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.
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
Offset: 0
Keywords
Examples
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)
Links
- 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.
Crossrefs
Programs
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Haskell
a026898 n = sum $ zipWith (^) [n + 1, n .. 1] [0 ..] -- Reinhard Zumkeller, Sep 14 2014
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Magma
[(&+[(n-k+1)^k: k in [0..n]]): n in [0..50]]; // Stefano Spezia, Jan 09 2019
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Maple
a:= n-> add((n+1-j)^j, j=0..n): seq(a(n), n=0..23); # Zerinvary Lajos, Apr 18 2009
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Mathematica
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), ", ")) -
Sage
[sum((n-j+1)^j for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 15 2021
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
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.
Extensions
a(23)-a(25) from Paul D. Hanna, Dec 28 2013
Comments