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 A008930 Number of compositions (p_1, p_2, p_3, ...) of n with 1 <= p_i <= i for all i. 36
 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) is the number of compositions (p_1,p_2,...) of n with 1<=p_i<=i for all i. a(n) is the 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) Roland Bacher, Generic numerical semigroups, hal-03221466 [math.CO], 2021. Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020. 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-François 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: A351972 A298118 A339292 * A339151 A164362 A329667 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 June 9 10:32 EDT 2023. Contains 363178 sequences. (Running on oeis4.)