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A141799
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Number of repeated integer partitions of n.
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3
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1, 3, 8, 25, 66, 192, 511, 1418, 3812, 10383, 27958, 75758, 204215, 551821, 1488561, 4018722, 10842422, 29262357, 78955472, 213063551, 574905487, 1551325859, 4185959285, 11295211039, 30478118079, 82240300045, 221911189754, 598790247900, 1615732588962
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OFFSET
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1,2
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COMMENTS
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An integer n can be partitioned into P(n) partitions P([n],i) where i=1,...,P(n) counts the partitions. The partition P([n],i) consists of T(n,i) integer parts t(i,j) with j=1,...,T(n,i). Now we perform on each t(i,j) an integer partition again and arrive at new partitions. Their parts can be partitioned again and so forth. We count such repeated partitions of n. One convention is necessary to avoid an infinite loop: The trivial partition P([n],1)=[n] will not be partitioned again but just counted once (and therefore we also have a(1)=1).
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LINKS
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FORMULA
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Let sum_{i=1}^P(n) denote the sum over all integer partitions P([n],i) of n. Let sum_{j=1}^T(i,j) denote the sum over all parts of the i-th integer partition. Then we have the recursive formula 1 if t(i,j)=n a(n) = sum_{i=1}^P(n) sum_{j=1}^T(i,j) { a(t(i,j)) else. E.g. a(4)=25 because [4] contributes 1, [1,3] contributes a(1)+a(3)=1+8=9, [2,2] contributes a(2)+a(2)=3+3=6, [1,1,2] contributes a(1)+a(1)+a(2)=1+1+3=5, [1,1,1,1] contributes a(1)+a(1)+a(1)+a(1)=1+1+1+1=4 which gives in total 25.
a(n) ~ c * d^n, where d = 2.69832910647421123126399866... (see A246828), c = 0.5088820425072641934222229579416714164592334575899644931509447692360546... . - Vaclav Kotesovec, Sep 04 2014
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EXAMPLE
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For the integers 1, 2, 3 and 4 we have
[1] -> 1,
thus a(1)=1.
[2] -> 1,
[1,1] => [1] ->, [1] -> 1.
thus a(2)=3.
[3] -> 1,
[1,2] => [1] -> 1, [2] -> 3,
[1,1,1] => [1] -> 1, [1] -> 1, [1] -> 1,
thus a(3)=8.
[4] -> 1,
[1,3] => [1] -> 1, [3] -> 8,
[2,2] => [2] -> 3, [2] -> 3,
[1,1,2] => [1] -> 1, [1] -> 1, [2] -> 3,
[1,1,1,1] => [1] -> 1, [1] -> 1, [1] -> 1, [1] -> 1,
thus a(4)=25.
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MAPLE
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A141799 := proc(n) option remember ; local a, P, i, p ; if n =1 then 1; else a := 0 ; for P in combinat[partition](n) do if nops(P) > 1 then for i in P do a := a+procname(i) ; od: else a := a+1 ; fi; od: RETURN(a) ; fi ; end: for n from 1 to 40 do printf("%d, ", A141799(n)) ; od: # R. J. Mathar, Aug 25 2008
# second Maple program
a:= proc(n) option remember;
1+ `if`(n>1, b(n, n-1)[2], 0)
end:
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, n]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
[f[1]+g[1], f[2]+g[2] +g[1]*a(i)]
fi
end:
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MATHEMATICA
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a[n_] := a[n] = 1 + If[n>1, b[n, n-1][[2]], 0]; b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*a[i]}]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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