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 A072706 Number of unimodal partitions/compositions of n into distinct terms. 5
 1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0). G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - Vladeta Jovovic, Jun 24 2003 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] EXAMPLE 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). From Joerg Arndt, Mar 25 2014: (Start) The a(10) = 33 such compositions of 10 are: 01:  [ 1 2 3 4 ] 02:  [ 1 2 4 3 ] 03:  [ 1 2 7 ] 04:  [ 1 3 4 2 ] 05:  [ 1 3 6 ] 06:  [ 1 4 3 2 ] 07:  [ 1 4 5 ] 08:  [ 1 5 4 ] 09:  [ 1 6 3 ] 10:  [ 1 7 2 ] 11:  [ 1 9 ] 12:  [ 2 3 4 1 ] 13:  [ 2 3 5 ] 14:  [ 2 4 3 1 ] 15:  [ 2 5 3 ] 16:  [ 2 7 1 ] 17:  [ 2 8 ] 18:  [ 3 4 2 1 ] 19:  [ 3 5 2 ] 20:  [ 3 6 1 ] 21:  [ 3 7 ] 22:  [ 4 3 2 1 ] 23:  [ 4 5 1 ] 24:  [ 4 6 ] 25:  [ 5 3 2 ] 26:  [ 5 4 1 ] 27:  [ 6 3 1 ] 28:  [ 6 4 ] 29:  [ 7 2 1 ] 30:  [ 7 3 ] 31:  [ 8 2 ] 32:  [ 9 1 ] 33:  [ 10 ] (End) MAPLE b:= proc(n, i) option remember; `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)))))     end: a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n\$2)): seq(a(n), n=0..100);  # Alois P. Heinz, Mar 25 2014 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A000009, A000041, A001523, A032020, A059618, A072705, A072707. Sequence in context: A091916 A102437 A319794 * A117433 A159284 A078028 Adjacent sequences:  A072703 A072704 A072705 * A072707 A072708 A072709 KEYWORD nonn AUTHOR Henry Bottomley, Jul 04 2002 STATUS approved

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Last modified December 10 16:44 EST 2018. Contains 318049 sequences. (Running on oeis4.)