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A057601 a(0) = a(1) = 1; a(n+1) is the number of partitions of n into parts a(k), 0 <= k <= n, each k occurring at most once.

%I A057601 #19 Jan 24 2024 17:08:03
%S A057601 1,1,2,2,4,4,6,8,10,11,12,14,17,21,25,29,33,37,43,49,54,59,66,72,80,
%T A057601 89,98,106,116,126,137,148,161,174,187,200,216,232,248,266,284,302,
%U A057601 321,344,367,391,414,440,465,493,523,556,584,616,650,689,726,768,808
%N A057601 a(0) = a(1) = 1; a(n+1) is the number of partitions of n into parts a(k), 0 <= k <= n, each k occurring at most once.
%H A057601 Alois P. Heinz, <a href="/A057601/b057601.txt">Table of n, a(n) for n = 0..10000</a>
%e A057601 a(6) = 6 because 5 = a(0) + a(4) = a(0) + a(5) = a(1) + a(4) = a(1) + a(5) = a(0) + a(2) + a(3) = a(1) + a(2) + a(3).
%p A057601 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
%p A057601        b(n, i-1)+`if`(a(i)>n, 0, b(n-a(i), i-1))))
%p A057601     end:
%p A057601 a:= proc(n) a(n):= `if`(n<2, 1, b(n-1, n-1)) end:
%p A057601 seq(a(n), n=0..70);  # _Alois P. Heinz_, May 26 2013
%t A057601 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 0, 0, b[n, i - 1] + If[a[i] > n, 0, b[n - a[i], i - 1]]]];
%t A057601 a[n_] := If[n < 2, 1, b[n - 1, n - 1]];
%t A057601 a /@ Range[0, 70] (* _Jean-François Alcover_, Nov 10 2020, after _Alois P. Heinz_ *)
%K A057601 nonn
%O A057601 0,3
%A A057601 _Leroy Quet_, Oct 06 2000
%E A057601 More terms from _Naohiro Nomoto_, Oct 28 2001
%E A057601 Extended beyond a(45) by _Alois P. Heinz_, May 26 2013