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 A072704 Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms. 23
 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 7, 5, 1, 1, 6, 12, 12, 9, 6, 1, 1, 7, 16, 20, 16, 11, 7, 1, 1, 8, 21, 30, 28, 20, 13, 8, 1, 1, 9, 27, 42, 45, 36, 24, 15, 9, 1, 1, 10, 33, 58, 68, 60, 44, 28, 17, 10, 1, 1, 11, 40, 77, 98, 95, 75, 52, 32, 19, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 REFERENCES Rigoberto Flórez, Leandro Junes, José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3094, Table 4. LINKS Alois P. Heinz, Rows n = 1..141, flattened H. Bottomley, Illustration of initial terms Eric Weisstein's World of Mathematics, Unimodal Sequence FORMULA G.f. with initial column 1, 0, 0, ...: 1 + Sum_{n>=1} (t*x^n / ( ( Product_{k=1..n-1} (1 - t*x^k)^2 ) * (1 - t*x^n) ) ). - Joerg Arndt, Oct 01 2017 EXAMPLE Rows start: 01:  [1] 02:  [1, 1] 03:  [1, 2, 1] 04:  [1, 3, 3, 1] 05:  [1, 4, 5, 4, 1] 06:  [1, 5, 8, 7, 5, 1] 07:  [1, 6, 12, 12, 9, 6, 1] 08:  [1, 7, 16, 20, 16, 11, 7, 1] 09:  [1, 8, 21, 30, 28, 20, 13, 8, 1] 10:  [1, 9, 27, 42, 45, 36, 24, 15, 9, 1] ... T(6,3)=8 since 6 can be written as 1+1+4, 1+2+3, 1+3+2, 1+4+1, 2+2+2, 2+3+1, 3+2+1, or 4+1+1 but not 2+1+3 or 3+1+2. MAPLE b:= proc(n, i) option remember; local q; `if`(i>n, 0,       `if`(irem(n, i, 'q')=0, x^q, 0) +expand(       add(b(n-i*j, i+1)*(j+1)*x^j, j=0..n/i)))     end: T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)): seq(T(n), n=1..12);  # Alois P. Heinz, Mar 26 2014 MATHEMATICA b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i ] == 0, x^Quotient[n, i], 0] + Expand[ Sum[b[n-i*j, i+1]*(j+1)*x^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *) unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], unimodQ]], {n, 0, 10}, {k, 0, n}] (* Gus Wiseman, Mar 06 2020 *) PROG (PARI) \\ starting for n=0, with initial column 1, 0, 0, ...: N=25;  x='x+O('x^N); T=Vec(1 + sum(n=1, N, t*x^n / ( prod(k=1, n-1, (1 - t*x^k)^2 ) * (1 - t*x^n) ) ) ) for(r=1, #T, print(Vecrev(T[r])) ); \\ Joerg Arndt, Oct 01 2017 CROSSREFS Cf. A059623, A072705. Row sums are A001523. First column is A057427, second is A000027 offset, third appears to be A000212 offset, right hand columns include A000012, A000027, A005408 and A008574. The case of partitions is A072233. Dominates A332670 (the version for negated compositions). The strict case is A072705. The case of constant compositions is A113704. Unimodal sequences covering an initial interval are A007052. Partitions whose run-lengths are unimodal are A332280. Cf. A107429, A115981, A227038, A328509, A332282, A332283, A332578, A332638, A332669, A332728. Sequence in context: A193515 A259874 A256141 * A038792 A196416 A329329 Adjacent sequences:  A072701 A072702 A072703 * A072705 A072706 A072707 KEYWORD nonn,tabl AUTHOR Henry Bottomley, Jul 04 2002 STATUS approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)