%I #26 Oct 05 2021 13:00:13

%S 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 513,665,859,1105,1416,1809,2306,2935,3731,4737,6005,7598,9593,12085,

%U 15192,19061,23875,29861,37299,46532,57978,72145,89650,111243,137837,170545,210725,260034,320492,394557,485213,596074,731508

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

%H Joerg Arndt and Alois P. Heinz, <a href="/A238874/b238874.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{n>=0} q^(n*(n+3)/2) / (1-q)^n. - _Joerg Arndt_, Mar 30 2014

%e The a(13) = 25 such composition of 13 are:

%e 01: [ 2 3 8 ]

%e 02: [ 2 4 7 ]

%e 03: [ 2 5 6 ]

%e 04: [ 2 6 5 ]

%e 05: [ 2 7 4 ]

%e 06: [ 2 11 ]

%e 07: [ 3 3 7 ]

%e 08: [ 3 4 6 ]

%e 09: [ 3 5 5 ]

%e 10: [ 3 6 4 ]

%e 11: [ 3 10 ]

%e 12: [ 4 3 6 ]

%e 13: [ 4 4 5 ]

%e 14: [ 4 5 4 ]

%e 15: [ 4 9 ]

%e 16: [ 5 3 5 ]

%e 17: [ 5 4 4 ]

%e 18: [ 5 8 ]

%e 19: [ 6 3 4 ]

%e 20: [ 6 7 ]

%e 21: [ 7 6 ]

%e 22: [ 8 5 ]

%e 23: [ 9 4 ]

%e 24: [ 10 3 ]

%e 25: [ 13 ]

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p add(b(n-j, i+1), j=i..n))

%p end:

%p a:= n-> b(n, 2):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Mar 24 2014

%t 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 (PARI) N=66; q='q+O('q^N);

%o gf=sum(n=0,N, q^(n*(n+3)/2) / (1-q)^n );

%o v=Vec(gf) \\ _Joerg Arndt_, Mar 30 2014

%Y Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).

%Y Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).

%Y Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).

%Y Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

%K nonn

%O 0,6

%A _Joerg Arndt_, Mar 23 2014