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A238861
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Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.
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9
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1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 18, 24, 32, 41, 55, 72, 95, 125, 164, 212, 275, 355, 459, 592, 763, 980, 1257, 1605, 2044, 2598, 3298, 4179, 5290, 6685, 8435, 10623, 13353, 16751, 20978, 26228, 32746, 40831, 50850, 63247, 78569, 97475, 120770, 149429, 184641, 227853, 280832, 345722, 425134, 522232, 640847, 785604
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: 1 + sum(n>=1, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) ). [Joerg Arndt, Mar 30 2014]
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EXAMPLE
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There are a(12) = 24 such compositions of 12:
01: [ 1 2 3 6 ]
02: [ 1 2 4 5 ]
03: [ 1 2 5 4 ]
04: [ 1 2 9 ]
05: [ 1 3 3 5 ]
06: [ 1 3 4 4 ]
07: [ 1 3 8 ]
08: [ 1 4 3 4 ]
09: [ 1 4 7 ]
10: [ 1 5 6 ]
11: [ 1 6 5 ]
12: [ 1 7 4 ]
13: [ 1 8 3 ]
14: [ 1 11 ]
15: [ 2 3 7 ]
16: [ 2 4 6 ]
17: [ 2 5 5 ]
18: [ 2 6 4 ]
19: [ 2 10 ]
20: [ 3 4 5 ]
21: [ 3 9 ]
22: [ 4 8 ]
23: [ 5 7 ]
24: [ 12 ]
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i=0, add(b(n-j, j+1), j=1..n),
add(b(n-j, i+1), j=i..n)))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j+1], {j, 1, n}], Sum[ b[n-j, i+1], {j, i, n}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
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PROG
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(PARI) N=66; q='q+O('q^N);
gf = 1 + sum(n=1, N, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) );
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CROSSREFS
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Cf. A238860 (partitions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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