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 A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k. (Formerly M2745) 23
 0, 1, 3, 8, 22, 65, 209, 732, 2780, 11377, 49863, 232768, 1151914, 6018785, 33087205, 190780212, 1150653920, 7241710929, 47454745803, 323154696184, 2282779990494, 16700904488705, 126356632390297, 987303454928972, 7957133905608836, 66071772829247409 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For n > 0: a(n) = sum of row n of triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014 a(n-1) is the number of set partitions of [n] into two or more blocks such that all absolute differences between least elements of consecutive blocks are 1. a(3) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, May 22 2017 Min_{n >= 1} a(n+1)/a(n) = 8/3. This is the answer to the 4th problem proposed during the first day of the final round of the 16th Austrian Mathematical Olympiad in 1985 (see link IMO Compendium). - Bernard Schott, Jan 07 2019 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Charles R Greathouse IV, Table of n, a(n) for n = 0..598 Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974. Mathematics StackExchange, Asymptotics of ..., 2011. Daniel Ropp, Problem 2 - 16th Austrian Mathematical Olympiad (Final round), Crux Mathematicorum, page 7, Vol. 14, Jun. 88. The IMO compendium, Problem 2, 16th Austrian Mathematical Olympiad, 1985. FORMULA G.f.: G(0)/x-1/(1-x)/x 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) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013 G.f.: Sum_{k>=1} x^k/(1 - (k + 1)*x). - Ilya Gutkovskiy, Oct 09 2018 a(n) = n^1 + (n-1)^2 + (n-2)^3 + ... + 3^(n-2) + 2^(n-1) + 1^n. - Bernard Schott, Jan 07 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 + w(n))) * w(n)^(n+2 - w(n)), where w(n) = (n+1)/LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics StackExchange link. EXAMPLE For n = 3 we get a(3) = 3^1 + 2^2 + 1^3 = 8. For n = 4 we get a(4) = 4^1 + 3^2 + 2^3 + 1^4 = 22. MAPLE A003101 := n->add((n-k+1)^k, k=1..n); P:=proc(n) local a, i, k; for i from 0 by 1 to n do k:=i; a:=0; while k>0 do a:=a+k^(i-k+1); k:=k-1; od; print(a); od; end: P(100); # Paolo P. Lava, Feb 29 2008 a:= n-> add((n-j+1)^j, j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Jun 07 2008 MATHEMATICA Table[Sum[(n-k+1)^k, {k, n}], {n, 0, 25}] (* Harvey P. Dale, Aug 14 2011 *) PROG (PARI) a(n)=sum(k=1, n, (n-k+1)^k) \\ Charles R Greathouse IV, Oct 31 2011 (Haskell) a003101 n = sum \$ zipWith (^) [0 ..] [n + 1, n .. 1] -- Reinhard Zumkeller, Sep 14 2014 CROSSREFS a(n) = A026898(n)-1. First differences are in A047970. Cf. A062810. Cf. A051129, A247358, A287215. Sequence in context: A099324 A290898 A117420 * A064443 A000732 A092090 Adjacent sequences:  A003098 A003099 A003100 * A003102 A003103 A003104 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 23 19:26 EDT 2021. Contains 348215 sequences. (Running on oeis4.)