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A238874 Strictly superdiagonal compositions: compositions (p1, p2, p3, ...) of n such that pi > i.

%I A238874 #26 Oct 05 2021 13:00:13
%S A238874 1,0,1,1,1,2,3,4,5,7,10,14,19,25,33,44,59,79,105,138,180,234,304,395,
%T A238874 513,665,859,1105,1416,1809,2306,2935,3731,4737,6005,7598,9593,12085,
%U A238874 15192,19061,23875,29861,37299,46532,57978,72145,89650,111243,137837,170545,210725,260034,320492,394557,485213,596074,731508
%N A238874 Strictly superdiagonal compositions: compositions (p1, p2, p3, ...) of n such that pi > i.
%H A238874 Joerg Arndt and Alois P. Heinz, <a href="/A238874/b238874.txt">Table of n, a(n) for n = 0..1000</a>
%F A238874 G.f.: Sum_{n>=0} q^(n*(n+3)/2) / (1-q)^n. - _Joerg Arndt_, Mar 30 2014
%e A238874 The a(13) = 25 such composition of 13 are:
%e A238874 01:  [ 2 3 8 ]
%e A238874 02:  [ 2 4 7 ]
%e A238874 03:  [ 2 5 6 ]
%e A238874 04:  [ 2 6 5 ]
%e A238874 05:  [ 2 7 4 ]
%e A238874 06:  [ 2 11 ]
%e A238874 07:  [ 3 3 7 ]
%e A238874 08:  [ 3 4 6 ]
%e A238874 09:  [ 3 5 5 ]
%e A238874 10:  [ 3 6 4 ]
%e A238874 11:  [ 3 10 ]
%e A238874 12:  [ 4 3 6 ]
%e A238874 13:  [ 4 4 5 ]
%e A238874 14:  [ 4 5 4 ]
%e A238874 15:  [ 4 9 ]
%e A238874 16:  [ 5 3 5 ]
%e A238874 17:  [ 5 4 4 ]
%e A238874 18:  [ 5 8 ]
%e A238874 19:  [ 6 3 4 ]
%e A238874 20:  [ 6 7 ]
%e A238874 21:  [ 7 6 ]
%e A238874 22:  [ 8 5 ]
%e A238874 23:  [ 9 4 ]
%e A238874 24:  [ 10 3 ]
%e A238874 25:  [ 13 ]
%p A238874 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238874       add(b(n-j, i+1), j=i..n))
%p A238874     end:
%p A238874 a:= n-> b(n, 2):
%p A238874 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 24 2014
%t A238874 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 2]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Mar 23 2015, after _Alois P. Heinz_ *)
%o A238874 (PARI) N=66; q='q+O('q^N);
%o A238874 gf=sum(n=0,N, q^(n*(n+3)/2) / (1-q)^n );
%o A238874 v=Vec(gf) \\ _Joerg Arndt_, Mar 30 2014
%Y A238874 Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).
%Y A238874 Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
%Y A238874 Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
%Y A238874 Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
%K A238874 nonn
%O A238874 0,6
%A A238874 _Joerg Arndt_, Mar 23 2014