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A008930 Number of compositions (p_1, p_2, p_3, ...) of n with 1 <= p_i <= i for all i. 17
1, 1, 1, 2, 3, 6, 11, 21, 41, 80, 157, 310, 614, 1218, 2421, 4819, 9602, 19147, 38204, 76266, 152307, 304256, 607941, 1214970, 2428482, 4854630, 9705518, 19405030, 38800412, 77585314, 155145677, 310251190, 620437691, 1240771141, 2481374234 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Previous name was: Number of increasing sequences of permutation type with maximal element n.

a(n)=number of compositions (p_1,p_2,...) of n with 1<=p_i<=i for all i. a(n)=number of Dyck n-paths with strictly increasing peaks. To get the correspondence, given such a Dyck path, split the path after the first up step reaching height i, i=1,2,...,h where h is the path's maximum height and count up steps in each block. Example: U-U-DUU-U-DDDD has been split as specified, yielding the composition (1,1,2,1). - David Callan, Feb 18 2004

Row sums of triangle A177517.

LINKS

Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)

M. Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications, vol.40, no.02 (April 2006), pp.107-121.

FORMULA

G.f.: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} (1-x^k)/(1-x). - Paul D. Hanna, Mar 20 2003

G.f.: A(x) = 1/(1 - x/(1+x) /(1 - x/(1+x+x^2) /(1 - x/(1+x+x^2+x^3) /(1 - x/(1+x+x^2+x^3+x^4) /(1 - x/(1+x+x^2+x^3+x^4+x^5) /(1 -...)))))), a continued fraction. - Paul D. Hanna, May 15 2012

limit(n->infinity) a(n+1)/a(n) = 2. - Mats Granvik, Feb 22 2011

a(n) ~ c * 2^(n-1), where c = 0.288788095086602421... (see constant A048651). - Vaclav Kotesovec, Mar 17 2014

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 21*x^7 +...

The g.f. equals the following series involving q-factorials:

A(x) = 1 + x + x^2*(1+x) + x^3*(1+x)*(1+x+x^2) + x^4*(1+x)*(1+x+x^2)*(1+x+x^2+x^3) + x^5*(1+x)*(1+x+x^2)*(1+x+x^2+x^3)*(1+x+x^2+x^3+x^4) +...

From Joerg Arndt, Dec 28 2012: (Start)

There are a(7)=21 compositions p(1)+p(2)+...+p(m)=7 such that p(k)<=k:

[ 1]  [ 1 1 1 1 1 1 1 ]

[ 2]  [ 1 1 1 1 1 2 ]

[ 3]  [ 1 1 1 1 2 1 ]

[ 4]  [ 1 1 1 1 3 ]

[ 5]  [ 1 1 1 2 1 1 ]

[ 6]  [ 1 1 1 2 2 ]

[ 7]  [ 1 1 1 3 1 ]

[ 8]  [ 1 1 1 4 ]

[ 9]  [ 1 1 2 1 1 1 ]

[10]  [ 1 1 2 1 2 ]

[11]  [ 1 1 2 2 1 ]

[12]  [ 1 1 2 3 ]

[13]  [ 1 1 3 1 1 ]

[14]  [ 1 1 3 2 ]

[15]  [ 1 2 1 1 1 1 ]

[16]  [ 1 2 1 1 2 ]

[17]  [ 1 2 1 2 1 ]

[18]  [ 1 2 1 3 ]

[19]  [ 1 2 2 1 1 ]

[20]  [ 1 2 2 2 ]

[21]  [ 1 2 3 1 ]

(End)

MAPLE

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

      add(b(n-j, i+1), j=1..min(i, n)))

    end:

a:= n-> b(n, 1):

seq(a(n), n=0..50);  # Alois P. Heinz, Mar 25 2014

MATHEMATICA

Sum[x^n*Product[(1-x^k)/(1-x), {k, 1, n}], {n, 0, 40}]+O[x]^41

Table[SeriesCoefficient[1+Sum[x^j*Product[(1-x^k)/(1-x), {k, 1, j}], {j, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Mar 17 2014 *)

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, i+1], {j, 1, Min[i, n]}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Apr 15 2015, after Alois P. Heinz *)

PROG

(PARI) { n=8; v=vector(n); for (i=1, n, v[i]=vector(i!)); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, for (a=0, i-1, v[i][j+a*k]=v[i-1][j]+a+1))); c=vector(n); for (i=1, n, for (j=1, i!, if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

(PARI) N=66; q='q+O('q^N); Vec( sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014

CROSSREFS

Cf. A048285, A177517.

Sequence in context: A006861 A052956 A298118 * A164362 A026742 A316471

Adjacent sequences:  A008927 A008928 A008929 * A008931 A008932 A008933

KEYWORD

nonn

AUTHOR

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it)

EXTENSIONS

More terms from Paul D. Hanna, Mar 20 2003

Corrected offset to 0, Joerg Arndt, Mar 24 2014

New name (using comment by David Callan) from Joerg Arndt, Mar 25 2014

STATUS

approved

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Last modified February 19 12:09 EST 2019. Contains 320310 sequences. (Running on oeis4.)