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A136405
Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i = n.
2
1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 4, 3, 7, 1, 4, 6, 7, 5, 11, 1, 3, 7, 8, 11, 7, 15, 1, 5, 8, 16, 14, 17, 11, 22, 1, 4, 12, 14, 23, 21, 25, 15, 30, 1, 6, 12, 24, 29, 38, 33, 37, 22, 42, 1, 5, 15, 24, 41, 42, 57, 47, 52, 30, 56, 1, 7, 18, 37, 47, 75, 68, 87, 70, 74, 42, 77
OFFSET
1,3
LINKS
FORMULA
T(n,1) = 1.
T(n,2) = A028242(n).
T(n,n) = A000041(n).
EXAMPLE
Triangle begins:
1;
1, 2;
1, 1, 3;
1, 3, 2, 5;
1, 2, 4, 3, 7;
1, 4, 6, 7, 5, 11;
1, 3, 7, 8, 11, 7, 15;
1, 5, 8, 16, 14, 17, 11, 22;
1, 4, 12, 14, 23, 21, 25, 15, 30;
1, 6, 12, 24, 29, 38, 33, 37, 22, 42;
...
T(4,2) = 3 since (4,2) can be bi-partitioned as (2,2) or ((1,1),(3,1)) or ((2,1),(2,1)).
T(5,3) = 4 since (5,3) can be bi-partitioned as ((1,1),(2,2)) or ((3,1),(1,2)) or ((1,1),(1,1),(3,1)) or ((1,1),(2,1),(2,1)).
PROG
(PARI)
P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, n, sum(k=0, n\w, (x*y)^(k*w)/P(k, w, n))), n)/y)}
{ for(n=1, 10, print(T(n))) } \\ Andrew Howroyd, Oct 23 2019
CROSSREFS
Row sums are A006171.
Sequence in context: A326408 A214717 A293312 * A210871 A308399 A373272
KEYWORD
nonn,tabl
AUTHOR
Benoit Jubin, Apr 13 2008
EXTENSIONS
Terms a(57) and beyond from Andrew Howroyd, Oct 23 2019
STATUS
approved