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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.
9
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
OFFSET
1,46
LINKS
FORMULA
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}.
T(n,k) = A015716(n,k) - A238450(n,k). - Andrew Howroyd, Apr 29 2020
EXAMPLE
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
PROG
(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
CROSSREFS
Columns k=1..6 are A238215, A238217, A238218, A238219, A238220, A238221.
Row sums are A238132.
Sequence in context: A037866 A333181 A306216 * A205777 A053398 A065833
KEYWORD
nonn,tabl
AUTHOR
Mircea Merca, Feb 26 2014
STATUS
approved