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A072706 Number of unimodal partitions/compositions of n into distinct terms.

%I A072706 #36 Dec 31 2019 16:38:47
%S A072706 1,1,1,3,3,5,9,11,15,21,33,39,55,69,93,127,159,201,261,327,411,537,
%T A072706 653,819,1011,1257,1529,1899,2331,2829,3441,4179,5031,6093,7305,8767,
%U A072706 10575,12573,14997,17847,21223,25089,29757,35055,41379,48801,57285,67131
%N A072706 Number of unimodal partitions/compositions of n into distinct terms.
%C A072706 Also the number of ways to partition a strict integer partition of n into two unordered blocks. - _Gus Wiseman_, Dec 31 2019
%H A072706 Alois P. Heinz, <a href="/A072706/b072706.txt">Table of n, a(n) for n = 0..1000</a>
%F A072706 a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0).
%F A072706 G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - _Vladeta Jovovic_, Jun 24 2003
%F A072706 G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [_Joerg Arndt_, Jan 20 2014]
%F A072706 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - _Vaclav Kotesovec_, Sep 22 2019
%e A072706 a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).
%e A072706 From _Joerg Arndt_, Mar 25 2014: (Start)
%e A072706 The a(10) = 33 such compositions of 10 are:
%e A072706 01:  [ 1 2 3 4 ]
%e A072706 02:  [ 1 2 4 3 ]
%e A072706 03:  [ 1 2 7 ]
%e A072706 04:  [ 1 3 4 2 ]
%e A072706 05:  [ 1 3 6 ]
%e A072706 06:  [ 1 4 3 2 ]
%e A072706 07:  [ 1 4 5 ]
%e A072706 08:  [ 1 5 4 ]
%e A072706 09:  [ 1 6 3 ]
%e A072706 10:  [ 1 7 2 ]
%e A072706 11:  [ 1 9 ]
%e A072706 12:  [ 2 3 4 1 ]
%e A072706 13:  [ 2 3 5 ]
%e A072706 14:  [ 2 4 3 1 ]
%e A072706 15:  [ 2 5 3 ]
%e A072706 16:  [ 2 7 1 ]
%e A072706 17:  [ 2 8 ]
%e A072706 18:  [ 3 4 2 1 ]
%e A072706 19:  [ 3 5 2 ]
%e A072706 20:  [ 3 6 1 ]
%e A072706 21:  [ 3 7 ]
%e A072706 22:  [ 4 3 2 1 ]
%e A072706 23:  [ 4 5 1 ]
%e A072706 24:  [ 4 6 ]
%e A072706 25:  [ 5 3 2 ]
%e A072706 26:  [ 5 4 1 ]
%e A072706 27:  [ 6 3 1 ]
%e A072706 28:  [ 6 4 ]
%e A072706 29:  [ 7 2 1 ]
%e A072706 30:  [ 7 3 ]
%e A072706 31:  [ 8 2 ]
%e A072706 32:  [ 9 1 ]
%e A072706 33:  [ 10 ]
%e A072706 (End)
%p A072706 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
%p A072706       expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
%p A072706     end:
%p A072706 a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):
%p A072706 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 25 2014
%t A072706 b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, Expand[b[n, i - 1] + If[i > n, 0, x*b[n - i, i - 1]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jan 16 2015, after _Alois P. Heinz_ *)
%t A072706 Table[If[n==0,1,Sum[2^(Length[ptn]-1),{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,15}] (* _Gus Wiseman_, Dec 31 2019 *)
%o A072706 (PARI) N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ _Joerg Arndt_, Mar 25 2014
%Y A072706 The non-strict version is A001523.
%Y A072706 Cf. A000009, A000041, A001970, A032020, A059618, A072705, A072707, A270995, A294617.
%K A072706 nonn
%O A072706 0,4
%A A072706 _Henry Bottomley_, Jul 04 2002