A238450 - OEIS

%I #16 Apr 29 2020 14:29:03

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

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

%U 3,3,2,2,2,1,1,0,0,1,4,4,3,3,2,2,2,2,1,1,0,0,1

%N Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

%H Andrew Howroyd, <a href="/A238450/b238450.txt">Table of n, a(n) for n = 1..1275</a>

%F T(n,k) = Sum_{j=1..round(n/(2*k))} A067661(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067659(n-2*j*k).

%F G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q)_{inf} + (1/2)*(q^k/(1-q^k))*(q;q)_{inf}.

%F T(n,k) = A015716(n,k) - A238451(n,k). - _Andrew Howroyd_, Apr 29 2020

%e n\k | 1 2 3 4 5 6 7 8 9 10

%e    1: 1

%e    2: 0 1

%e    3: 0 0 1

%e    4: 0 0 0 1

%e    5: 0 0 0 0 1

%e    6: 1 1 1 0 0 1

%e    7: 1 1 0 1 0 0 1

%e    8: 2 1 1 1 1 0 0 1

%e    9: 2 2 2 1 1 1 0 0 1

%e   10: 3 2 2 1 2 1 1 0 0 1

%o (PARI) T(n,k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) + prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, m==0)} \\ _Andrew Howroyd_, Apr 29 2020

%Y Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.

%Y Row sums are A238131.

%Y Cf. A015716, A067659, A067661, A238451.

%K nonn,tabl

%O 1,29

%A _Mircea Merca_, Feb 26 2014

%E Terms a(79) and beyond from _Andrew Howroyd_, Apr 29 2020