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A090764
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Number of partitions of n with two sorts of part 1.
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4
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1, 2, 5, 11, 24, 50, 104, 212, 431, 870, 1752, 3518, 7057, 14138, 28310, 56661, 113377, 226820, 453728, 907561, 1815259, 3630683, 7261576, 14523405, 29047130, 58094643, 116189764, 232380102, 464760912, 929522671, 1859046381, 3718094000, 7436189507, 14872380808
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OFFSET
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0,2
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COMMENTS
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Original name was: a(n) = Sum_{pi = partition of n} 2^{number of 1's in pi}.
a(n) is the number of compositions of n consisting of two kinds of parts, p and p', when the order of all the primed parts does not matter; or equivalently, when the order of all the unprimed parts does not matter. - Gregory L. Simay, Sep 12 2017
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LINKS
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FORMULA
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Convolution of A000041 with A011782. In general, Sum_{pi = partition of n} k^{number of 1's in pi} is equal to the convolution of the partitions of n with the compositions of n having parts of (k-1) kinds; this is k=2. - Gregory L. Simay, Sep 15 2017
a(n) ~ c * 2^n, where c = Product_{n>=2} (2^n/(2^n-1)) = 1.7313733097275318... - Vaclav Kotesovec, Sep 17 2017
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EXAMPLE
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a(4) = 24 because the partitions of 4 are 4(1), 31(2), 22(1), 211(4) and 1111(16). 1+2+1+4+16=24.
a(4) = 24 because the compositions of 4 (when the parts are of two kinds, p and p', and the order of the primed parts does not matter) are 4; 4'; 3,1; 1,3; 3',1 = 1,3'; 3,1' = 1',3; 3'1' = 1'3'; 2,2; 2'2 = 2,2'; 2',2'; 2,1,1; 1,2,1; 1,1,2; 2,1,1'= 2,1',1 = 1',2,1; 2',1,1 = 1,2',1 = 1,1,2'; 2,1',1' = 1',2,1' = 1',1',2; 2',1',1 = 2',1,1'= 1,2',1' = 1',2',1 = 1',1,2' = 1,1',2'; 2',1',1' = 1',2',1' = 1',1',2'; 1,1,1,1; 1',1,1,1 = 1,1',1,1 = 1,1,1',1 = 1,1,1,1'; 1',1',1,1 = 1,1',1,1' = 1',1,1',1 = 1',1,1,1' = 1,1'1',1 = 1,1,1',1'; 1',1',1',1 = 1',1',1,1' = 1',1,1',1', 1,1',1',1'; 1',1',1',1'. - Gregory L. Simay, Sep 12 2017
a(4) = 24 because the convolution of the first 5 partition numbers with the first 5 composition numbers is 1*8 + 1*4 + 2*2 + 3*1 + 5*1 = 24. (Note that the first partition number is A000041(0)=1; and the first composition number is A011782(0)=1.) - Gregory L. Simay, Sep 15 2017
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
add(b(n-i*j, i-1), j=0..iquo(n, i)))
end:
a:= n-> b(n, n):
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MATHEMATICA
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c[n_] := Count[n, 1]; f[n_] := Apply[Plus, 2^Map[ c, IntegerPartitions[n] ]]; Table[ f[n], {n, 0, 31}] (* Robert G. Wilson v, Feb 12 2004 *)
b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, Sum[b[n - i*j, i - 1], {j, 0, Quotient[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
Table[PartitionsP[n] + Sum[2^(k-1)*PartitionsP[n-k], {k, 1, n}], {n, 0, 50}] (* Vaclav Kotesovec, Apr 10 2017 *)
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PROG
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(Java) import java.math.*; import java.io.*; public class A090764 { public static final int LIMIT = 80; public static final BigInteger TWO = new BigInteger("2"); public static void main(String[] args) throws Exception {BigInteger[] a = new BigInteger[LIMIT];
int i, j; PrintStream out = new PrintStream(new FileOutputStream("A090764.txt")); a[0] = BigInteger.ONE; for (i = 1; i < LIMIT; i++)a[i] = a[i - 1].multiply(TWO); for (j = 2; j < LIMIT; j++)for (i = j; i < LIMIT; i++)
a[i] = a[i - 1].multiply(TWO); for (j = 2; j < LIMIT; j++)for (i = j; i < LIMIT; i++) a[i] = a[i].add(a[i - j]); for (i = 0; i < LIMIT; i++)out.print(a[i] + " "); out.print(" "); }} // David Wasserman, Feb 10 2004
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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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