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A028310 Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x. 91

%I #132 Jan 05 2024 08:01:34

%S 1,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,

%T 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,

%U 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71

%N Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.

%C 1 followed by the natural numbers.

%C Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).

%C Engel expansion of e (see A006784 for definition) [when offset by 1]. - _Henry Bottomley_, Dec 18 2000

%C Also the denominators of the series expansion of log(1+x). Numerators are A062157. - _Robert G. Wilson v_, Aug 14 2015

%C The right-shifted sequence (with a(0)=0) is an autosequence (of the first kind - see definition in links). - _Jean-François Alcover_, Mar 14 2017

%H G. C. Greubel, <a href="/A028310/b028310.txt">Table of n, a(n) for n = 0..10000</a>

%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).

%H Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, <a href="https://arxiv.org/abs/2306.03155">Pattern-avoiding stabilized-interval-free permutations</a>, arXiv:2306.03155 [math.CO], 2023.

%H Olivia Nabawanda and Fanja Rakotondrajao, <a href="https://arxiv.org/abs/2011.07304">The sets of flattened partitions with forbidden patterns</a>, arXiv:2011.07304 [math.CO], 2020.

%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

%H Oeis Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>

%H E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (<a href="http://neilsloane.com/doc/self.txt">Abstract</a>, <a href="http://neilsloane.com/doc/self.pdf">pdf</a>, <a href="http://neilsloane.com/doc/self.ps">ps</a>).

%H Michael Somos, <a href="http://cis.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%F Binomial transform is A005183. - _Paul Barry_, Jul 21 2003

%F G.f.: (1 - x + x^2) / (1 - x)^2 = (1 - x^6) /((1 - x) * (1 - x^2) * (1 - x^3)) = (1 + x^3) / ((1 - x) * (1 - x^2)). a(0) = 1, a(n) = n if n>0.

%F Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - _Michael Somos_ Jul 30 2006

%F G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - _Michael Somos_, Apr 05 2012

%F G.f. of A112934(x) = 1 / (1 - a(0)*x / (1 - a(1)*x / ...)). - _Michael Somos_, Apr 05 2012

%F a(n) = A000027(n) unless n=0.

%F a(n) = Sum_{k=0..n} A123110(n,k). - _Philippe Deléham_, Oct 06 2009

%F E.g.f: 1+x*exp(x). - _Wolfdieter Lang_, May 03 2010

%F a(n) = sqrt(floor[A204503(n+3)/9]). - _M. F. Hasler_, Jan 16 2012

%F E.g.f.: 1-x + x*E(0), where E(k) = 2 + x/(2*k+1 - x/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 24 2013

%F a(n) = A001477(n) + A000007(n). - _Miko Labalan_, Dec 12 2015 (See the first comment.)

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9 + ...

%p a:= n-> `if`(n=0, 1, n):

%p seq(a(n), n=0..60);

%t Denominator@ CoefficientList[Series[Log[1+x], {x,0,75}], x] (* or *)

%t CoefficientList[ Series[(1 -x +x^2)/(1-x)^2, {x,0,75}], x] (* _Robert G. Wilson v_, Aug 14 2015 *)

%t Join[{1}, Range[75]] (* _G. C. Greubel_, Jan 05 2024 *)

%o (PARI) {a(n) = (n==0) + max(n, 0)} /* _Michael Somos_, Feb 02 2004 */

%o (PARI) A028310(n)=n+!n \\ _M. F. Hasler_, Jan 16 2012

%o (Haskell)

%o a028310 n = 0 ^ n + n

%o a028310_list = 1 : [1..] -- _Reinhard Zumkeller_, Nov 06 2012

%o (Python)

%o def A028310(n): return n|bool(n)^1 # _Chai Wah Wu_, Jul 13 2023

%o (Magma) [n eq 0 select 1 else n: n in [0..75]]; // _G. C. Greubel_, Jan 05 2024

%o (SageMath) [n + int(n==0) for n in range(76)] # _G. C. Greubel_, Jan 05 2024

%Y Cf. A000007, A000027, A000660 (boustrophedon transform).

%Y Cf. A001477, A004001, A005229, A112934, A123110, A212393, A204503.

%K nonn,easy,mult

%O 0,3

%A _N. J. A. Sloane_

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)