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.
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, 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, 3, 3, 2, 2, 2, 1, 1, 0, 0, 1, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1
Offset: 1
Examples
n\k | 1 2 3 4 5 6 7 8 9 10 1: 1 2: 0 1 3: 0 0 1 4: 0 0 0 1 5: 0 0 0 0 1 6: 1 1 1 0 0 1 7: 1 1 0 1 0 0 1 8: 2 1 1 1 1 0 0 1 9: 2 2 2 1 1 1 0 0 1 10: 3 2 2 1 2 1 1 0 0 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
Crossrefs
Programs
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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
Formula
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).
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}.
Extensions
Terms a(79) and beyond from Andrew Howroyd, Apr 29 2020