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%I #23 Jan 09 2023 07:41:28
%S 1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,
%T 1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,
%U 0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0
%N Array read by rows: T(n,k) = number of partitions of n into distinct odd parts in which k is the greatest part, for k=1,2,...,n, n>=1.
%C First T(n,k) not 0 or 1 is T(17,9)=2, which counts 1+7+9 and 3+5+9. Row sums: A000700.
%H Alois P. Heinz, <a href="/A027356/b027356.txt">Rows n = 1..361, flattened</a>
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a027/A027356.java">Java program</a> (github)
%F T(n, 1)=0 for all n; T(n, n)=1 for all odd n>1; and for n>=3, T(n, k)=0 if k is even, else T(n, k)=Sum{T(n-k, i): i=1, 2, ..., n-1} for k=2, 3, ..., n-1.
%e First 5 rows:
%e 1
%e 0 0
%e 0 0 1
%e 0 0 1 0
%e 0 0 0 0 1
%e Row 40 with even-numbered terms deleted:
%e 0 0 0 0 0 0 2 5 6 7 6 5 4 3 2 1 1 1 1;
%e E.g. final 2 counts these two partitions: 9+31 and 1+3+5+31.
%p b:= proc(n, i) option remember; `if`(n>i^2, 0, `if`(n=0, 1,
%p b(n, i-1) +(p-> `if`(p>n, 0, b(n-p, i-1)))((2*i-1))))
%p end:
%p T:= (n, k)-> `if`(k::even, 0, b(n-k, (k-1)/2)):
%p seq(seq(T(n, k), k=1..n), n=1..20); # _Alois P. Heinz_, Oct 28 2019
%t b[n_, i_] := b[n, i] = If[n > i^2, 0, If[n == 0, 1, b[n, i - 1] + Function[p, If[p > n, 0, b[n - p, i - 1]]][2i - 1]]];
%t T [n_, k_] := If[EvenQ[k], 0, b[n - k, (k - 1)/2]];
%t Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)
%Y Cf. A000700.
%Y T(4n+1,2n+1) gives A069910.
%K nonn,tabl
%O 1,145
%A _Clark Kimberling_, revised Jul 23 2004
%E Edited by _N. J. A. Sloane_, Sep 14 2008 at the suggestion of _R. J. Mathar_
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