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%I #13 Jan 02 2023 21:55:17
%S 0,0,1,2,5,9,18,35,67,131,257,505,996,1973,3915,7781,15486,30855,
%T 61527,122764,245069,489412,977673,1953515,3904108,7803545,15599618,
%U 31187269,62355347,124679883,249310255,498540890,996953659,1993701032,3987069747,7973603891
%N Number of integer compositions of n with exactly one part above the diagonal.
%H Andrew Howroyd, <a href="/A351983/b351983.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(2) = 1 through a(6) = 18 compositions:
%e (2) (3) (4) (5) (6)
%e (21) (13) (14) (15)
%e (22) (32) (42)
%e (31) (41) (51)
%e (211) (131) (114)
%e (212) (132)
%e (221) (141)
%e (311) (213)
%e (2111) (222)
%e (312)
%e (321)
%e (411)
%e (1311)
%e (2112)
%e (2121)
%e (2211)
%e (3111)
%e (21111)
%t pless[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],pless[#]==1&]],{n,0,10}]
%o (PARI)
%o S(v,u,c=0)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
%o seq(n)={my(v=vector(1+n), s=0); v[1]=1; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1)), O(x^2)); s+=apply(p->polcoef(p,1), v)); s} \\ _Andrew Howroyd_, Jan 02 2023
%Y The version for permutations is A000295, weak A057427.
%Y The version for partitions is A002620, weak A001477.
%Y The weak version is A177510.
%Y The version for fixed points is A240736, nonfixed A352520.
%Y This is column k = 1 of A352524; column k = 0 is A008930.
%Y A238349 counts compositions by fixed points, first column A238351.
%Y A352521 counts compositions by strong nonexcedances, first column A219282.
%Y A352522 counts compositions by weak nonexcedances, first column A238874.
%Y A352523 counts compositions by nonfixed points, first column A010054.
%Y A352524 counts compositions by strong excedances, first column A008930.
%Y A352525 counts compositions by weak excedances, first column A177510.
%Y Cf. A088218, A098825, A115994, A238352, A330644, A352516.
%K nonn
%O 0,4
%A _Gus Wiseman_, Apr 02 2022
%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2023
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