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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.
9
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
1,29
LINKS
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}.
T(n,k) = A015716(n,k) - A238451(n,k). - Andrew Howroyd, Apr 29 2020
EXAMPLE
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
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, m==0)} \\ Andrew Howroyd, Apr 29 2020
CROSSREFS
Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.
Row sums are A238131.
Sequence in context: A229745 A339366 A016397 * A251926 A335504 A037908
KEYWORD
nonn,tabl
AUTHOR
Mircea Merca, Feb 26 2014
EXTENSIONS
Terms a(79) and beyond from Andrew Howroyd, Apr 29 2020
STATUS
approved