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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

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

Examples

			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  + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• 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).
• Michael Somos, Rational Function Multiplicative Coefficients
• Index entries for linear recurrences with constant coefficients, signature (2,-1).
• Index entries for Molien series
• Index entries for sequences related to Engel expansions

Crossrefs

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

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

Programs

Haskell

a028310 n = 0 ^ n + n
a028310_list = 1 : [1..]  -- Reinhard Zumkeller, Nov 06 2012

Magma

[n eq 0 select 1 else n: n in [0..75]]; // G. C. Greubel, Jan 05 2024

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 *)
LinearRecurrence[{2,-1},{1,1,2},80] (* Harvey P. Dale, Jan 29 2025 *)

PARI

{a(n) = (n==0) + max(n, 0)} /* Michael Somos, Feb 02 2004 */

PARI

A028310(n)=n+!n  \\ M. F. Hasler, Jan 16 2012

Python

def A028310(n): return n|bool(n)^1 # Chai Wah Wu, Jul 13 2023

SageMath

[n + int(n==0) for n in range(76)] # G. C. Greubel, Jan 05 2024

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.)