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A126796
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Number of complete partitions of n.
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1
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1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 31, 39, 55, 71, 100, 125, 173, 218, 291, 366, 483, 600, 784, 971, 1244, 1538, 1957, 2395, 3023, 3693, 4605, 5604, 6942, 8397, 10347, 12471, 15235, 18309, 22267, 26619, 32219, 38414, 46216, 54941, 65838
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| A partition of n is complete if every number 1 to n can be represented as a sum of parts of the partition. This generalizes perfect partitions, where the representation for each number must be unique.
A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Mar 22 2007
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REFERENCES
| SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.
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LINKS
| David W. Wilson, Table of n, a(n) for n = 0..300
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EXAMPLE
| a(5) = 4 because there are 4 complete partitions of 5: {1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 2, 2} and {1, 1, 3}.
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MAPLE
| isCompl := proc(p, n) local m, pers, reps, f, lst, s; reps := {}; pers := combinat[permute](p); for m from 1 to nops(pers) do lst := op(m, pers); for f from 1 to nops(lst) do s := add( op(i, lst), i=1..f); reps := reps union {s}; od; od; for m from 1 to n do if not m in reps then RETURN(false); fi; od; RETURN(true); end: A126796 := proc(n) local prts, res, p; prts := combinat[partition](n); res :=0; for p from 1 to nops(prts) do if isCompl(op(p, prts), n) then res := res+1; fi; od; RETURN(res); end: for n from 1 to 40 do printf("%d %d ", n, A126796(n)); od; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 27 2007
At the beginning of the 2nd Maple program replace the current 15 by any other positive integer n in order to obtain a(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007
with(combinat): a:=proc(n) local P, b, k, p, S, j: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i], i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: end: seq(a(n), n=1..30); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007
with(combinat): n:=15: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i], i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: b; - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007
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PROG
| (PARI) T(n, k)=if(k<=1, 1, if(n<2*k-1, T(n, floor((n+1)/2)), T(n, k-1)+T(n-k, k))) a(n)=T(n, floor((n+1)/2)) /* If modified to save earlier results, this would be efficient. */ - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Mar 22 2007
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CROSSREFS
| Cf. A002033.
Sequence in context: A027337 A193146 A027193 * A206138 A157162 A109434
Adjacent sequences: A126793 A126794 A126795 * A126797 A126798 A126799
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KEYWORD
| nonn
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AUTHOR
| Brian Hopkins (bhopkins(AT)spc.edu), Feb 20 2007
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 27 2007
More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007
Further terms from Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) and David W. Wilson, Mar 22 2007
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