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A238861 Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.

%I A238861 #20 Sep 20 2019 02:17:51
%S A238861 1,1,1,2,2,3,4,6,7,10,13,18,24,32,41,55,72,95,125,164,212,275,355,459,
%T A238861 592,763,980,1257,1605,2044,2598,3298,4179,5290,6685,8435,10623,13353,
%U A238861 16751,20978,26228,32746,40831,50850,63247,78569,97475,120770,149429,184641,227853,280832,345722,425134,522232,640847,785604
%N A238861 Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.
%H A238861 Vaclav Kotesovec, <a href="/A238861/b238861.txt">Table of n, a(n) for n = 0..8000</a> (terms 0..1000 from Alois P. Heinz)
%F A238861 G.f.: 1 + sum(n>=1, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) ). [_Joerg Arndt_, Mar 30 2014]
%e A238861 There are a(12) = 24 such compositions of 12:
%e A238861 01:  [ 1 2 3 6 ]
%e A238861 02:  [ 1 2 4 5 ]
%e A238861 03:  [ 1 2 5 4 ]
%e A238861 04:  [ 1 2 9 ]
%e A238861 05:  [ 1 3 3 5 ]
%e A238861 06:  [ 1 3 4 4 ]
%e A238861 07:  [ 1 3 8 ]
%e A238861 08:  [ 1 4 3 4 ]
%e A238861 09:  [ 1 4 7 ]
%e A238861 10:  [ 1 5 6 ]
%e A238861 11:  [ 1 6 5 ]
%e A238861 12:  [ 1 7 4 ]
%e A238861 13:  [ 1 8 3 ]
%e A238861 14:  [ 1 11 ]
%e A238861 15:  [ 2 3 7 ]
%e A238861 16:  [ 2 4 6 ]
%e A238861 17:  [ 2 5 5 ]
%e A238861 18:  [ 2 6 4 ]
%e A238861 19:  [ 2 10 ]
%e A238861 20:  [ 3 4 5 ]
%e A238861 21:  [ 3 9 ]
%e A238861 22:  [ 4 8 ]
%e A238861 23:  [ 5 7 ]
%e A238861 24:  [ 12 ]
%p A238861 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238861       `if`(i=0, add(b(n-j, j+1), j=1..n),
%p A238861        add(b(n-j, i+1), j=i..n)))
%p A238861     end:
%p A238861 a:= n-> b(n, 0):
%p A238861 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 26 2014
%t A238861 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, i, n}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)
%o A238861 (PARI) N=66; q='q+O('q^N);
%o A238861 gf = 1 + sum(n=1, N, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) );
%o A238861 v=Vec(gf) \\ _Joerg Arndt_, Mar 30 2014
%Y A238861 Cf. A238860 (partitions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
%Y A238861 Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
%Y A238861 Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
%Y A238861 Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
%K A238861 nonn
%O A238861 0,4
%A A238861 _Joerg Arndt_, Mar 24 2014