cp's OEIS Frontend

A238450 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.

%I A238450 #16 Apr 29 2020 14:29:03
%S A238450 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 A238450 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 A238450 3,3,2,2,2,1,1,0,0,1,4,4,3,3,2,2,2,2,1,1,0,0,1
%N A238450 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 A238450 Andrew Howroyd, <a href="/A238450/b238450.txt">Table of n, a(n) for n = 1..1275</a>
%F A238450 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 A238450 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 A238450 T(n,k) = A015716(n,k) - A238451(n,k). - _Andrew Howroyd_, Apr 29 2020
%e A238450 n\k | 1 2 3 4 5 6 7 8 9 10
%e A238450    1: 1
%e A238450    2: 0 1
%e A238450    3: 0 0 1
%e A238450    4: 0 0 0 1
%e A238450    5: 0 0 0 0 1
%e A238450    6: 1 1 1 0 0 1
%e A238450    7: 1 1 0 1 0 0 1
%e A238450    8: 2 1 1 1 1 0 0 1
%e A238450    9: 2 2 2 1 1 1 0 0 1
%e A238450   10: 3 2 2 1 2 1 1 0 0 1
%o A238450 (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 A238450 Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.
%Y A238450 Row sums are A238131.
%Y A238450 Cf. A015716, A067659, A067661, A238451.
%K A238450 nonn,tabl
%O A238450 1,29
%A A238450 _Mircea Merca_, Feb 26 2014
%E A238450 Terms a(79) and beyond from _Andrew Howroyd_, Apr 29 2020