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A088567
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Number of "non-squashing" partitions of n into distinct parts.
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17
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1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19, 24, 25, 31, 32, 40, 41, 50, 51, 63, 64, 77, 78, 95, 96, 114, 115, 138, 139, 163, 164, 194, 195, 226, 227, 266, 267, 307, 308, 357, 358, 408, 409, 471, 472, 535, 536, 612, 613, 690, 691, 785, 786, 881, 882, 995, 996, 1110, 1111
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OFFSET
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0,4
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COMMENTS
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"Non-squashing" refers to the property that p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k: if the parts are stacked in increasing size, at no point does the sum of the parts above a certain part exceed the size of that part.
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LINKS
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FORMULA
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a(0)=1, a(1)=1; and for m >= 1, a(2m) = a(2m-1) + a(m) - 1, a(2m+1) = a(2m) + 1.
Or, a(0)=1, a(1)=1; and for m >= 1, a(2m) = a(0)+a(1)+...+a(m)-1; a(2m+1) = a(0)+a(1)+...+a(m).
G.f.: 1 + x/(1-x) + Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1-x^(2^j)).
G.f.: Product_{n>=0} 1/(1-x^(2^n)) - Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ). (The two terms correspond to A000123 and A088931 respectively.)
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EXAMPLE
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The partitions of n = 1 through 6 are: 1; 2; 3=1+2; 4=1+3; 5=1+4=2+3; 6=1+5=2+4=1+2+3.
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MAPLE
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f := proc(n) option remember; local t1, i; if n <= 2 then RETURN(1); fi; t1 := add(f(i), i=0..floor(n/2)); if n mod 2 = 0 then RETURN(t1-1); fi; t1; end;
t1 := 1 + x/(1-x); t2 := add( x^(3*2^(k-1))/ mul( (1-x^(2^j)), j=0..k), k=1..10); series(t1+t2, x, 256); # increase 10 to get more terms
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MATHEMATICA
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max = 63; f = 1 + x/(1-x) + Sum[x^(3*2^(k-1))/Product[(1-x^(2^j)), {j, 0, k}], {k, 1, Log[2, max]}]; s = Series[f, {x, 0, max}] // Normal; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 06 2014 *)
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PROG
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(Haskell)
import Data.List (transpose)
a088567 n = a088567_list !! n
a088567_list = 1 : tail xs where
xs = 0 : 1 : zipWith (+) xs (tail $ concat $ transpose [xs, tail xs])
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CROSSREFS
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Cf. A187821 (non-squashing partitions of n into odd 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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