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A325547
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Number of compositions of n with strictly increasing differences.
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12
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1, 1, 2, 3, 6, 8, 11, 18, 24, 30, 45, 57, 71, 96, 120, 148, 192, 235, 286, 354, 431, 518, 628, 752, 893, 1063, 1262, 1482, 1744, 2046, 2386, 2775, 3231, 3733, 4305, 4977, 5715, 6536, 7507, 8559, 9735, 11112, 12608, 14252, 16177, 18265, 20553, 23204, 26090, 29223
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OFFSET
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0,3
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
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LINKS
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EXAMPLE
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The a(1) = 1 through a(6) = 11 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(31) (32) (33)
(112) (41) (42)
(211) (113) (51)
(212) (114)
(311) (213)
(312)
(411)
(2112)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@Differences[#]&]], {n, 0, 15}]
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PROG
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(PARI) \\ Row sums of R(n) give A179269 (breakdown by width)
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L, v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, v[i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M, n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j, w]))); x^i*(1 + x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 27 2019
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CROSSREFS
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Cf. A000079, A000740, A008965, A034297, A070211, A175342, A179269, A179254, A240027, A325545, A325546, A325548, A325552, A325557.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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