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A000065
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-1 + number of partitions of n.
(Formerly M1012 N0379)
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22
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0, 0, 1, 2, 4, 6, 10, 14, 21, 29, 41, 55, 76, 100, 134, 175, 230, 296, 384, 489, 626, 791, 1001, 1254, 1574, 1957, 2435, 3009, 3717, 4564, 5603, 6841, 8348, 10142, 12309, 14882, 17976, 21636, 26014, 31184, 37337, 44582, 53173, 63260, 75174, 89133
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n+1) is the number of noncongruent n-dimensional integer-sided simplices with diameter n. - Sascha Kurz (sascha.kurz(AT)uni-byreuth.de), Jul 26 2004
Also, the number of partitions of n into parts each less than n.
Also, the number of distinct types of equation which can be derived from the equation [n,0,0] not including itself. (Ince)
Also, the number of rooted trees on n nodes with height exactly 2.
Also, the number of partitions (of any positive integer) whose sum + length is <= n. Example: a(5) = 6 counts 4, 3, 21, 2, 11, 1. Proof: Given a partition of n other than the all 1s partition, subtract 1 from each part and then drop the zeros. This is a bijection to the partitions with sum + length <= n. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007
a(n) = A026820(n,n-1) for n>1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010]
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REFERENCES
| E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, p. 498; MR0010757.
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n=0..199
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MAPLE
| with (combstruct):ZL:=proc(m) local i; [T0, {seq(T.i=Prod(Z, Set(T.(i+1))), i=0..m-1), T.m=Z}, unlabeled] end:A:=n -> count(ZL(2), size=n)-count(ZL(1), size=n): seq(A(n), n=1..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 05 2007
ZL :=[S, {S = Set(Cycle(Z), 1 < card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2008
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(1/eta(x+x*O(x^n)), n)-1)
(PARI) a(n)=if(n<0, 0, numbpart(n)-1)
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CROSSREFS
| A000041 - 1. A diagonal of A058716.
Sequence in context: A103259 A082380 A136460 * A023499 A103445 A001747
Adjacent sequences: A000062 A000063 A000064 * A000066 A000067 A000068
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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