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A026807
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Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.
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23
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1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 4, 2, 1, 1, 1, 1, 22, 7, 3, 2, 1, 1, 1, 1, 30, 8, 4, 2, 1, 1, 1, 1, 1, 42, 12, 5, 3, 2, 1, 1, 1, 1, 1, 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1, 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1, 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 135, 34, 13
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.
T(n,g) is also the number of not necessarily connected 2-regular graphs with girth at least g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012
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LINKS
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Alois P. Heinz, Rows n = 1..141, flattened
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
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FORMULA
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G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic, Jun 22 2003
T(n, k) = T(n, k+1)+T(n-k, k) (where T(n, n) = 1). - Franklin T. Adams-Watters, Jan 24 2005
Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson, Jan 31 2008
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EXAMPLE
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Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) =
y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...
Triangle starts: - Jason Kimberley, Feb 05 2012
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 2, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 4, 2, 1, 1, 1, 1;
22, 7, 3, 2, 1, 1, 1, 1;
30, 8, 4, 2, 1, 1, 1, 1, 1;
42, 12, 5, 3, 2, 1, 1, 1, 1, 1;
56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;
77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;
101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
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MAPLE
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T:= proc(n, k) option remember;
`if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))
end:
seq (seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 28 2012
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PROG
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(Haskell)
import Data.List (tails)
a026807 n k = a026807_tabl !! (n-1) !! (k-1)
a026807_row n = a026807_tabl !! (n-1)
a026807_tabl = map
(\row -> map (p $ last row) $ init $ tails row) a002260_tabl
where p 0 _ = 1
p _ [] = 0
p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 01 2012
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CROSSREFS
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Row sums give A046746.
Cf. A026835.
Cf. A026794.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: this sequence (triangle); columns of this sequence: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular simple graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012
Cf. A002260.
Sequence in context: A210765 A160183 A168534 * A179045 A106740 A178534
Adjacent sequences: A026804 A026805 A026806 * A026808 A026809 A026810
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling
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STATUS
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approved
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