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%I #79 Feb 16 2018 20:24:27
%S 1,2,5,11,24,50,104,212,431,870,1752,3518,7057,14138,28310,56661,
%T 113377,226820,453728,907561,1815259,3630683,7261576,14523405,
%U 29047130,58094643,116189764,232380102,464760912,929522671,1859046381,3718094000,7436189507,14872380808
%N Number of partitions of n with two sorts of part 1.
%C Original name was: a(n) = Sum_{pi = partition of n} 2^{number of 1's in pi}.
%C 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
%H Alois P. Heinz, <a href="/A090764/b090764.txt">Table of n, a(n) for n = 0..1000</a>
%H Steve Butler, Jeongyoon Choi, Kimyung Kim, Kyuhyeok Seo, <a href="https://arxiv.org/abs/1702.05808">Enumerating multiplex juggling patterns</a>, arXiv:1702.05808 [math.CO], 2017.
%F G.f.: (1/(1-2*x))*Product_{m>=2} 1/(1-x^m). - _Vladeta Jovovic_, Feb 04 2004
%F 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
%F a(n) = 2*a(n-1) + A000041(n) - A000041(n-1). - _Gregory L. Simay_, Sep 16 2017
%F a(n) ~ c * 2^n, where c = Product_{n>=2} (2^n/(2^n-1)) = 1.7313733097275318... - _Vaclav Kotesovec_, Sep 17 2017
%e 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.
%e 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
%e 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
%e a(4) = 2*a(3) + A000041(4) - A000041(3) = 2*11 + 5 - 3 = 24. - _Gregory L. Simay_, Sep 16 2017
%p b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
%p add(b(n-i*j, i-1), j=0..iquo(n, i)))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Feb 19 2013
%t 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 *)
%t 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_ *)
%t Table[PartitionsP[n] + Sum[2^(k-1)*PartitionsP[n-k], {k, 1, n}], {n, 0, 50}] (* _Vaclav Kotesovec_, Apr 10 2017 *)
%o (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];
%o 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++)
%o 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
%Y Cf. A000041, A011782, A259401.
%Y Column k=2 of A292741.
%K nonn
%O 0,2
%A _Jon Perry_, Feb 01 2004
%E More terms from _David Wasserman_, Feb 10 2004
%E a(0) inserted by _Alois P. Heinz_, Feb 19 2013
%E New name from _Alois P. Heinz_, Sep 21 2017
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