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A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).

%I A059618 #28 Mar 26 2014 14:41:44
%S A059618 1,1,1,3,4,6,10,15,21,30,43,59,82,111,148,199,263,344,451,584,751,965,
%T A059618 1230,1560,1973,2483,3110,3885,4834,5990,7405,9123,11202,13724,16762,
%U A059618 20417,24815,30081,36377,43900,52860,63511,76166,91157,108886,129842
%N A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).
%H A059618 Alois P. Heinz, <a href="/A059618/b059618.txt">Table of n, a(n) for n = 0..1000</a>
%H A059618 R. C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/unimodal.pdf">Strongly Unimodal Sequences and Mixed Mock Modular Forms</a>
%F A059618 a(n) = A059619(n,0) = Sum_k A059619(n,k) for k>0 when n>0.
%F A059618 G.f.: sum(k>=0, x^k * prod(i=1..k-1, 1 + x^i)^2 ). - _Vladeta Jovovic_, Dec 05 2003
%e A059618 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).
%e A059618 From _Joerg Arndt_, Dec 09 2012: (Start)
%e A059618 The a(7) = 15 strongly unimodal compositions of 7 are
%e A059618 [ #]   composition
%e A059618 [ 1]   [ 1 2 3 1 ]
%e A059618 [ 2]   [ 1 2 4 ]
%e A059618 [ 3]   [ 1 3 2 1 ]
%e A059618 [ 4]   [ 1 4 2 ]
%e A059618 [ 5]   [ 1 5 1 ]
%e A059618 [ 6]   [ 1 6 ]
%e A059618 [ 7]   [ 2 3 2 ]
%e A059618 [ 8]   [ 2 4 1 ]
%e A059618 [ 9]   [ 2 5 ]
%e A059618 [10]   [ 3 4 ]
%e A059618 [11]   [ 4 2 1 ]
%e A059618 [12]   [ 4 3 ]
%e A059618 [13]   [ 5 2 ]
%e A059618 [14]   [ 6 1 ]
%e A059618 [15]   [ 7 ]
%e A059618 (End)
%p A059618 b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
%p A059618       `if`(n=0, 1, add(b(n-j, j, 0), j=1..min(n, i-1))+
%p A059618       `if`(t=1, add(b(n-j, j, 1), j=i+1..n), 0)))
%p A059618     end:
%p A059618 a:= n-> b(n, 0, 1):
%p A059618 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 21 2014
%t A059618 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_ *)
%o A059618 (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
%Y A059618 Cf. A000009, A000041, A001523, A059607, A059619.
%K A059618 nice,nonn
%O A059618 0,4
%A A059618 _Henry Bottomley_, Jan 31 2001