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A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

%I A238451 #16 Apr 29 2020 14:29:46
%S A238451 0,0,0,1,1,0,1,0,1,0,1,1,1,1,0,1,1,0,1,1,0,1,1,1,1,1,1,0,1,1,1,0,1,1,
%T A238451 1,0,1,1,1,1,1,1,1,1,0,2,2,2,2,0,1,1,1,1,0,2,2,2,1,2,1,1,1,1,1,0,3,3,
%U A238451 2,2,2,1,1,1,1,1,1,0,4,3,3,3,2,2,2,1,1,1,1,1,0
%N A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.
%H A238451 Andrew Howroyd, <a href="/A238451/b238451.txt">Table of n, a(n) for n = 1..1275</a>
%F A238451 T(n,k) = Sum_{j=1..round(n/(2*k))} A067659(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067661(n-2*j*k).
%F A238451 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 A238451 T(n,k) = A015716(n,k) - A238450(n,k). - _Andrew Howroyd_, Apr 29 2020
%e A238451 n/k | 1 2 3 4 5 6 7 8 9 10
%e A238451    1: 0
%e A238451    2: 0 0
%e A238451    3: 1 1 0
%e A238451    4: 1 0 1 0
%e A238451    5: 1 1 1 1 0
%e A238451    6: 1 1 0 1 1 0
%e A238451    7: 1 1 1 1 1 1 0
%e A238451    8: 1 1 1 0 1 1 1 0
%e A238451    9: 1 1 1 1 1 1 1 1 0
%e A238451   10: 2 2 2 2 0 1 1 1 1 0
%o A238451 (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, 0)} \\ _Andrew Howroyd_, Apr 29 2020
%Y A238451 Columns k=1..6 are A238215, A238217, A238218, A238219, A238220, A238221.
%Y A238451 Row sums are A238132.
%Y A238451 Cf. A015716, A067659, A067661, A238450.
%K A238451 nonn,tabl
%O A238451 1,46
%A A238451 _Mircea Merca_, Feb 26 2014