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 A238089 Number of compositions of n such that the minimum part is equal to 1 and the first 1 occurs before any maximum part in the composition. 1
 0, 0, 0, 1, 3, 7, 16, 34, 73, 152, 318, 657, 1356, 2784, 5703, 11647, 23740, 48290, 98073, 198887, 402837, 815032, 1647424, 3327112, 6714352, 13540995, 27292175, 54978561, 110697947, 222790629, 448211668, 901392243, 1812185325, 3642182184, 7318157714 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS We note that the definition implies that the maximum part of the composition must be strictly greater than 1. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..800 FORMULA a(n) = Sum_{i>=2} x^(i+1)/(1-Sum_{j=2..i-1} x^j)/(1 - Sum{k=1..i-1} x^k)/(1 - Sum_{m=1..i} x^m). a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014 EXAMPLE a(5) = 7 because we have: 1+4, 1+1+3, 1+2+2, 1+3+1, 1+1+1+2, 1+1+2+1, 1+2+1+1. MAPLE b:= proc(n, t, m) option remember;       `if`(n=0, t, add(`if`(j=1 and m>1, b(n-1, 1, m),       `if`(j>=m, b(n-j, 0, j), b(n-j, t, m))), j=1..n))     end: a:= n-> b(n, 0\$2): seq(a(n), n=0..45);  # Alois P. Heinz, Feb 17 2014 MATHEMATICA nn=30; CoefficientList[Series[Sum[x^(i+1)/(1-Sum[x^j, {j, 2, i-1}])/(1-Sum[x^k, {k, 1, i-1}])/(1-Sum[x^m, {m, 1, i}]), {i, 2, nn}], {x, 0, nn}], x] (* or *) Table[Length[Select[Level[Table[Select[Compositions[n, k], Count[#, 0]==0&], {k, 1, n}], {2}], Min[Position[#, 1]]

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Last modified December 6 13:05 EST 2021. Contains 349563 sequences. (Running on oeis4.)