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A246721
Number of partitions of n into parts of the n-th list of distinct parts in the order given by A246688.
3
1, 1, 1, 2, 0, 2, 0, 2, 2, 0, 14, 3, 4, 0, 20, 3, 2, 1, 0, 26, 24, 4, 4, 2, 1, 35, 31, 4, 24, 2, 6, 1, 0, 378, 54, 42, 42, 5, 31, 0, 2, 0, 0, 631, 78, 61, 56, 5, 45, 34, 3, 3, 2, 2, 0, 1045, 992, 110, 85, 75, 73, 6, 55, 0, 7, 42, 8, 0, 2, 0, 1772, 1581, 156
OFFSET
0,4
COMMENTS
The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .
LINKS
FORMULA
a(n) = A246720(n,n).
EXAMPLE
a(7) = 2 because there are 2 partitions of 7 into parts 1, 4: [1,1,1,1,1,1,1], [1,1,1,4].
MAPLE
b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
[map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
end:
f:= proc() local i, l; i, l:=0, [];
proc(n) while n>=nops(l)
do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
end
end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
end:
a:= n-> g(n, f(n)):
seq(a(n), n=0..80);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
a[n_] := g[n, f[n]];
a /@ Range[0, 80] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
CROSSREFS
Main diagonal of A246720.
Cf. A246688, A246691 (the same for compositions).
Sequence in context: A204173 A103668 A276812 * A249441 A076472 A161840
KEYWORD
nonn,look
AUTHOR
Alois P. Heinz, Sep 02 2014
STATUS
approved