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A238451
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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.
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9
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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, 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, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 0
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OFFSET
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1,46
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LINKS
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FORMULA
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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).
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}.
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EXAMPLE
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n/k | 1 2 3 4 5 6 7 8 9 10
1: 0
2: 0 0
3: 1 1 0
4: 1 0 1 0
5: 1 1 1 1 0
6: 1 1 0 1 1 0
7: 1 1 1 1 1 1 0
8: 1 1 1 0 1 1 1 0
9: 1 1 1 1 1 1 1 1 0
10: 2 2 2 2 0 1 1 1 1 0
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PROG
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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, 0)} \\ Andrew Howroyd, Apr 29 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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