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A129921 Number of generalized compositions of n: words b_1^{i_1}b_2^{i_2}...b_k^{i_k} such that b_j's and j_i's are positive integers and sum b_j*i_j = n. 13
1, 1, 3, 7, 18, 43, 108, 263, 651, 1599, 3942, 9698, 23890, 58805, 144806, 356512, 877820, 2161285, 5321485, 13102246, 32259890, 79428762, 195566238, 481514453, 1185564348, 2919044646, 7187145712, 17695877607, 43570023304, 107276219947, 264130857268, 650331536681, 1601218102939 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

If the additional constraint was added that b_j does not equal to b_{j+1}, the sequence generated would be the compositions (ordered partitions) of integers.

This is a variant of compositions of compositions: for each composition of n, write it in value^repetition form, and then choose a composition for each repetition factor. - Franklin T. Adams-Watters, May 27 2010.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Sylvie Corteel, PaweĊ‚ Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8

FORMULA

G.f.: 1/(1-sum(k>0, z^k/(1-z^k))

G.f.: 1/(1-sum(k>0, tau(k) x^k)), where tau(k) is the number of divisors of k. - Franklin T. Adams-Watters, May 27 2010.

G.f.: 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^k)^(1/k))). - Ilya Gutkovskiy, Oct 18 2018

EXAMPLE

a(3)=7 because we can write

3^{1},

1^{2} 2^{1},

2^{1} 1^{1},

1^{3},

1^{2} 1^{1},

1^{1} 1^{2},

1^{1} 1^{1} 1^{1}.

MAPLE

a:= proc(n) option remember; `if`(n=0, 1,

      add(add(a(n-i*j), j=1..n/i), i=1..n))

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, Jul 22 2017

MATHEMATICA

nmax = 50; CoefficientList[Series[1/(1 - Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2017 *)

PROG

(PARI) N=66;  A=vector(66);  A[0+1]=1;

for (n=1, N-1, A[n+1] = sum(k=0, n-1, A[k+1]*sigma(n-k, 0)) );

A  /* Joerg Arndt, Apr 28 2013 */

(PARI) N = 66;  x = 'x + O('x^N);

gf = 1/(1-sum(k=1, N, x^k/(1-x^k)) );

Vec(gf)  /* Joerg Arndt, Apr 28 2013 */

CROSSREFS

Cf. A000079, A011782, A000005.

Sequence in context: A000633 A036669 A091621 * A036670 A262321 A182995

Adjacent sequences:  A129918 A129919 A129920 * A129922 A129923 A129924

KEYWORD

nonn

AUTHOR

Pawel Hitczenko (phitczenko(AT)math.drexel.edu), Jun 05 2007

EXTENSIONS

Edited by Franklin T. Adams-Watters, May 27 2010

Added more terms, Joerg Arndt, Apr 28 2013

STATUS

approved

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Last modified August 19 20:20 EDT 2019. Contains 326133 sequences. (Running on oeis4.)