A156869 Triangle read by rows: T(n,k) = number of nondecreasing sequences of n positive integers with reciprocals adding up to k (1 <= k <= n).

1, 1, 1, 3, 1, 1, 14, 4, 1, 1, 147, 17, 4, 1, 1, 3462, 164, 18, 4, 1, 1, 294314, 3627, 167, 18, 4, 1, 1, 159330691, 297976, 3644, 168, 18, 4, 1, 1

Offset: 1

Author

Jens Voß, Feb 17 2009

Keywords

• more
• nonn
• tabl

Comments

Conjecture: T(2n + m, n + m) = T(2n, n) ( = A156870(n) ) if and only if m >= 0.

Yes, the diagonals are constant for n <= 2k. Any such sequence must have at least one 1; remove that 1, and you get a sequence for n-1,k-1. - Franklin T. Adams-Watters, Feb 20 2009

The next term will be a(37) = A002966(9). - M. F. Hasler, Feb 20 2009

Examples

			Triangle begins:
n=1:      1
n=2:      1,    1
n=3:      3,    1,   1
n=4:     14,    4,   1,  1
n=5:    147,   17,   4,  1, 1
n=6:   3462,  164,  18,  4, 1, 1
n=7: 294314, 3627, 167, 18, 4, 1, 1
For n = 4 and k = 2, the T(4, 2) = 4 sequences are (1, 2, 3, 6), (1, 2, 4, 4), (1, 3, 3, 3) and (2, 2, 2, 2) because 1/1 + 1/2 + 1/3 + 1/6 = 1/1 + 1/2 + 1/4 + 1/4 = 1/1 + 1/3 + 1/3 + 1/3 = 1/2 + 1/2 + 1/2 + 1/2 = 2.
		

Crossrefs

Cf. A002966 (column k=1), A156871 (row sums), A280519, A280520.

T(2n, n) = A156870(n).

Programs

PARI

{ A156869(n,k,m=1) = n==1 & return(numerator(k)==1 & denominator(k)>=m); sum( i=max(m,1\k+1),n\k, A156869(n-1, k-1/i, i)); } \\ M. F. Hasler, Feb 20 2009

Extensions

a(21)-a(36) from M. F. Hasler, Feb 20 2009