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A028310 Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x. 91
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, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
1 followed by the natural numbers.
Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).
Engel expansion of e (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000
Also the denominators of the series expansion of log(1+x). Numerators are A062157. - Robert G. Wilson v, Aug 14 2015
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
LINKS
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Olivia Nabawanda and Fanja Rakotondrajao, The sets of flattened partitions with forbidden patterns, arXiv:2011.07304 [math.CO], 2020.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Oeis Wiki, Autosequence
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
FORMULA
Binomial transform is A005183. - Paul Barry, Jul 21 2003
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.
Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - Michael Somos Jul 30 2006
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Apr 05 2012
G.f. of A112934(x) = 1 / (1 - a(0)*x / (1 - a(1)*x / ...)). - Michael Somos, Apr 05 2012
a(n) = A000027(n) unless n=0.
a(n) = Sum_{k=0..n} A123110(n,k). - Philippe Deléham, Oct 06 2009
E.g.f: 1+x*exp(x). - Wolfdieter Lang, May 03 2010
a(n) = sqrt(floor[A204503(n+3)/9]). - M. F. Hasler, Jan 16 2012
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
a(n) = A001477(n) + A000007(n). - Miko Labalan, Dec 12 2015 (See the first comment.)
EXAMPLE
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 + ...
MAPLE
a:= n-> `if`(n=0, 1, n):
seq(a(n), n=0..60);
MATHEMATICA
Denominator@ CoefficientList[Series[Log[1+x], {x, 0, 75}], x] (* or *)
CoefficientList[ Series[(1 -x +x^2)/(1-x)^2, {x, 0, 75}], x] (* Robert G. Wilson v, Aug 14 2015 *)
Join[{1}, Range[75]] (* G. C. Greubel, Jan 05 2024 *)
PROG
(PARI) {a(n) = (n==0) + max(n, 0)} /* Michael Somos, Feb 02 2004 */
(PARI) A028310(n)=n+!n \\ M. F. Hasler, Jan 16 2012
(Haskell)
a028310 n = 0 ^ n + n
a028310_list = 1 : [1..] -- Reinhard Zumkeller, Nov 06 2012
(Python)
def A028310(n): return n|bool(n)^1 # Chai Wah Wu, Jul 13 2023
(Magma) [n eq 0 select 1 else n: n in [0..75]]; // G. C. Greubel, Jan 05 2024
(SageMath) [n + int(n==0) for n in range(76)] # G. C. Greubel, Jan 05 2024
CROSSREFS
Cf. A000007, A000027, A000660 (boustrophedon transform).
Sequence in context: A088480 A289207 A061019 * A097045 A118760 A130446
KEYWORD
nonn,easy,mult
AUTHOR
STATUS
approved

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Last modified February 20 20:29 EST 2024. Contains 370217 sequences. (Running on oeis4.)