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.

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

Offset: 0

Author

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

Keywords

• nonn

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

INVERT transform of tau (A000005). - Alois P. Heinz, Feb 11 2021

Examples

			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}.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Sylvie Corteel and Paweł Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8.

Crossrefs

Cf. A000005, A000079, A011782, A353005.

Programs

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
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[tau](i), i=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 11 2021

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 *)

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 */

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

a(n) = Sum_{k=0..n-1} tau(n-k)*a(k) for n>0 with a(0) = 1. - Ridouane Oudra, Mar 13 2020

Extensions

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

More terms from Joerg Arndt, Apr 28 2013