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A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.
(Formerly M2745)
21
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

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.

R. K. Hoeflin, Mega Test [broken link]

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

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

N. J. A. Sloane, Henry W. Gould

STATUS

approved

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Last modified November 18 01:20 EST 2018. Contains 317279 sequences. (Running on oeis4.)