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 A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing). 10
 1, 1, 1, 3, 4, 6, 10, 15, 21, 30, 43, 59, 82, 111, 148, 199, 263, 344, 451, 584, 751, 965, 1230, 1560, 1973, 2483, 3110, 3885, 4834, 5990, 7405, 9123, 11202, 13724, 16762, 20417, 24815, 30081, 36377, 43900, 52860, 63511, 76166, 91157, 108886, 129842 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms FORMULA a(n) = A059619(n,0) = Sum_k A059619(n,k) for k>0 when n>0. G.f.: sum(k>=0, x^k * prod(i=1..k-1, 1 + x^i)^2 ). - Vladeta Jovovic, Dec 05 2003 EXAMPLE a(6) = 10 since 6 can be written as 6, 5+1, 4+2, 3+2+1, 2+4, 2+3+1, 1+5, 1+4+1, 1+3+2 or 1+2+3 (but for example neither 2+2+1+1 nor 1+2+2+1 which are only weakly unimodal). From Joerg Arndt, Dec 09 2012: (Start) The a(7) = 15 strongly unimodal compositions of 7 are [ #]   composition [ 1]   [ 1 2 3 1 ] [ 2]   [ 1 2 4 ] [ 3]   [ 1 3 2 1 ] [ 4]   [ 1 4 2 ] [ 5]   [ 1 5 1 ] [ 6]   [ 1 6 ] [ 7]   [ 2 3 2 ] [ 8]   [ 2 4 1 ] [ 9]   [ 2 5 ] [10]   [ 3 4 ] [11]   [ 4 2 1 ] [12]   [ 4 3 ] [13]   [ 5 2 ] [14]   [ 6 1 ] [15]   [ 7 ] (End) MAPLE b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,       `if`(n=0, 1, add(b(n-j, j, 0), j=1..min(n, i-1))+       `if`(t=1, add(b(n-j, j, 1), j=i+1..n), 0)))     end: a:= n-> b(n, 0, 1): seq(a(n), n=0..60);  # Alois P. Heinz, Mar 21 2014 MATHEMATICA s[n_?Positive, k_] := s[n, k] = Sum[s[n - k, j], {j, 0, k - 1}]; s[0, 0] = 1; s[0, _] = 0; s[_?Negative, _] = 0; t[n_, k_] := t[n, k] = s[n, k] + Sum[t[n - k, j], {j, k + 1, n}]; a[n_] := t[n, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Dec 06 2012, after Vladeta Jovovic *) PROG (PARI) N=66; x='x+O('x^N); Vec(sum(n=0, N, x^(n) * prod(k=1, n-1, 1+x^k)^2)) \\ Joerg Arndt, Mar 26 2014 CROSSREFS Cf. A000009, A000041, A001523, A059607, A059619. Sequence in context: A255879 A171096 A125869 * A114736 A099417 A139463 Adjacent sequences:  A059615 A059616 A059617 * A059619 A059620 A059621 KEYWORD nice,nonn AUTHOR Henry Bottomley, Jan 31 2001 STATUS approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)