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A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals. 3

%I

%S 1,1,0,1,1,0,1,0,1,0,1,1,1,1,0,1,0,2,0,1,0,1,1,0,2,1,1,0,1,0,1,1,3,0,

%T 1,0,1,1,0,2,0,3,1,1,0,1,0,1,0,2,0,4,0,1,0,1,0,1,1,1,2,1,4,1,1,0,1,1,

%U 0,1,2,0,3,0,5,0,1,0,1,1,2,0,1,2,0,3,0,5,1,1,0

%N Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%C 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], ... .

%H Alois P. Heinz, <a href="/A246720/b246720.txt">Antidiagonals n = 0..140, flattened</a>

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, ...

%e 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, ...

%e 0, 1, 0, 2, 1, 2, 0, 1, 1, 0, 3, 1, 0, 0, 2, ...

%e 0, 1, 1, 3, 0, 2, 1, 2, 1, 0, 4, 1, 2, 0, 4, ...

%e 0, 1, 0, 3, 0, 2, 0, 2, 1, 1, 5, 2, 0, 0, 4, ...

%e 0, 1, 1, 4, 1, 3, 0, 2, 2, 0, 7, 2, 2, 1, 6, ...

%e 0, 1, 0, 4, 0, 3, 0, 2, 1, 0, 8, 2, 0, 0, 6, ...

%e 0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0, 9, ...

%e 0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0, 9, ...

%e 0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

%p b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],

%p [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))

%p end:

%p f:= proc() local i, l; i, l:=0, [];

%p proc(n) while n>=nops(l)

%p do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]

%p end

%p end():

%p g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,

%p add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))

%p end:

%p A:= (n, k)-> g(n, f(k)):

%p seq(seq(A(n, d-n), n=0..d), d=0..16);

%Y Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.

%Y Main diagonal gives A246721.

%Y Cf. A246688, A246690 (the same for compositions).

%K nonn,tabl

%O 0,18

%A _Alois P. Heinz_, Sep 02 2014

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Last modified September 26 02:39 EDT 2020. Contains 337346 sequences. (Running on oeis4.)