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A238859 Compositions with subdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i.

%I A238859 #15 Mar 17 2024 10:04:54
%S A238859 1,1,2,4,7,14,26,51,99,195,383,759,1504,2988,5944,11840,23602,47084,
%T A238859 93975,187647,374812,748857,1496487,2991017,5978900,11952780,23897506,
%U A238859 47782081,95543378,191053334,382052880,764019152,1527898772,3055572646,6110782652,12220980359
%N A238859 Compositions with subdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i.
%H A238859 Alois P. Heinz, <a href="/A238859/b238859.txt">Table of n, a(n) for n = 0..1000</a>
%F A238859 a(n) ~ c * 2^n, where c = 1/2 - QPochhammer(1/2)/2 = 0.3556059524566987893605501390353846099555440475796571079426294669... - _Vaclav Kotesovec_, May 01 2014, updated Mar 17 2024
%e A238859 There are a(6) = 26 such compositions of 6:
%e A238859 01:  [ 1 1 1 1 1 1 ]
%e A238859 02:  [ 1 1 1 1 2 ]
%e A238859 03:  [ 1 1 1 2 1 ]
%e A238859 04:  [ 1 1 1 3 ]
%e A238859 05:  [ 1 1 2 1 1 ]
%e A238859 06:  [ 1 1 2 2 ]
%e A238859 07:  [ 1 1 3 1 ]
%e A238859 08:  [ 1 2 1 1 1 ]
%e A238859 09:  [ 1 2 1 2 ]
%e A238859 10:  [ 1 2 2 1 ]
%e A238859 11:  [ 1 2 3 ]
%e A238859 12:  [ 2 1 1 1 1 ]
%e A238859 13:  [ 2 1 1 2 ]
%e A238859 14:  [ 2 1 2 1 ]
%e A238859 15:  [ 2 1 3 ]
%e A238859 16:  [ 2 2 1 1 ]
%e A238859 17:  [ 2 2 2 ]
%e A238859 18:  [ 2 3 1 ]
%e A238859 19:  [ 3 1 1 1 ]
%e A238859 20:  [ 3 1 2 ]
%e A238859 21:  [ 3 2 1 ]
%e A238859 22:  [ 3 3 ]
%e A238859 23:  [ 4 1 1 ]
%e A238859 24:  [ 4 2 ]
%e A238859 25:  [ 5 1 ]
%e A238859 26:  [ 6 ]
%p A238859 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238859       `if`(i=0, add(b(n-j, j+1), j=1..n),
%p A238859        add(b(n-j, i+1), j=1..min(n,i))))
%p A238859     end:
%p A238859 a:= n-> b(n, 0):
%p A238859 seq(a(n), n=0..40);  # _Alois P. Heinz_, Mar 25 2014
%t A238859 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j+1], {j, 1, n}], Sum[ b[n-j, i+1], {j, 1, Min[n, i]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 13 2015, after _Alois P. Heinz_ *)
%Y A238859 Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
%Y A238859 Cf. A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
%Y A238859 Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
%Y A238859 Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
%K A238859 nonn
%O A238859 0,3
%A A238859 _Joerg Arndt_, Mar 24 2014