OFFSET
0,4
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 191.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..4100 (first 501 terms from Christian G. Bower)
L. Carlitz, Restricted Compositions, Fibonacci Quarterly, 14 (1976) 254-264.
Sylvie Corteel, Paweł Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 42 and 117.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 201
F. Harary & R. W. Robinson, The number of achiral trees, Jnl. Reine Angewandte Mathematik 278 (1975), 322-335. (Annotated scanned copy)
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Chapter 8.
FORMULA
a(n) = Sum_{k=1..n} A048272(k)*a(n-k), n>1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/(1 - Sum_{k>0} x^k/(1+x^k)).
a(n) ~ c r^n where c is approximately 0.456387 and r is approximately 1.750243. (Formula from Knopfmacher and Prodinger reference.) - Franklin T. Adams-Watters, May 27 2010. With better precision: r = 1.7502412917183090312497386246398158787782058181381590561316586... (see A241902), c = 0.4563634740588133495321001859298593318027266156100046548066205... - Vaclav Kotesovec, Apr 30 2014
G.f. is the special case p=2 of 1/(1 - Sum_{k>0} (z^k/(1-z^k) - p*z^(k*p)/(1-z^(k*p)))), see A129922. - Joerg Arndt, Apr 28 2013
G.f.: 1/(1 - x * (d/dx) log(Product_{k>=1} (1 + x^k)^(1/k))). - Ilya Gutkovskiy, Oct 18 2018
Moebius transform of A329738. - Gus Wiseman, Nov 27 2019
EXAMPLE
From Joerg Arndt, Oct 27 2012: (Start)
The 23 such compositions of n=7 are
[ 1] 1 2 1 2 1
[ 2] 1 2 1 3
[ 3] 1 2 3 1
[ 4] 1 2 4
[ 5] 1 3 1 2
[ 6] 1 3 2 1
[ 7] 1 4 2
[ 8] 1 5 1
[ 9] 1 6
[10] 2 1 3 1
[11] 2 1 4
[12] 2 3 2
[13] 2 4 1
[14] 2 5
[15] 3 1 2 1
[16] 3 1 3
[17] 3 4
[18] 4 1 2
[19] 4 2 1
[20] 4 3
[21] 5 2
[22] 6 1
[23] 7
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
add(`if`(j=i, 0, b(n-j, `if`(j<=n-j, j, 0))), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 27 2014
MATHEMATICA
A048272[n_] := Total[If[OddQ[#], 1, -1]& /@ Divisors[n]]; a[n_] := a[n] = Sum[A048272[k]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 38}](* Jean-François Alcover, Nov 25 2011, after Vladeta Jovovic *)
nmax = 50; CoefficientList[Series[1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 07 2020 *)
Table[Count[Flatten[Permutations/@IntegerPartitions[n], 1], _?(FreeQ[Differences[#], 0]&)], {n, 0, 20}] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Nov 23 2024 *)
PROG
(PARI) N = 66; x = 'x + O('x^N); p=2;
gf = 1/(1-sum(k=1, N, x^k/(1-x^k)-p*x^(k*p)/(1-x^(k*p))));
Vec(gf) /* Joerg Arndt, Apr 28 2013 */
(Haskell)
a003242 n = a003242_list !! n
a003242_list = 1 : f [1] where
f xs = y : f (y : xs) where
y = sum $ zipWith (*) xs a048272_list
-- Reinhard Zumkeller, Nov 04 2015
CROSSREFS
KEYWORD
nonn,nice,changed
AUTHOR
E. Rodney Canfield
EXTENSIONS
More terms from David W. Wilson
STATUS
approved