A121775 T(n, k) = Sum_{d|n} phi(n/d)*binomial(d,k) for n>0, T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

1, 1, 1, 2, 3, 1, 3, 5, 3, 1, 4, 8, 7, 4, 1, 5, 9, 10, 10, 5, 1, 6, 15, 20, 21, 15, 6, 1, 7, 13, 21, 35, 35, 21, 7, 1, 8, 20, 36, 60, 71, 56, 28, 8, 1, 9, 21, 42, 86, 126, 126, 84, 36, 9, 1, 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1, 11, 21, 55, 165, 330, 462, 462, 330, 165, 55

Offset

0,4

Comments

For n>0, (1/n)*Sum_{k=0..n} T(n,k)*(c-1)^k is the number of n-bead necklaces with c colors. See the cross references.

Links

Table of n, a(n) for n=0..75.

Example

Triangle begins:
[ 0]  1;
[ 1]  1,  1;
[ 2]  2,  3,  1;
[ 3]  3,  5,  3,   1;
[ 4]  4,  8,  7,   4,   1;
[ 5]  5,  9, 10,  10,   5,   1;
[ 6]  6, 15, 20,  21,  15,   6,   1;
[ 7]  7, 13, 21,  35,  35,  21,   7,   1;
[ 8]  8, 20, 36,  60,  71,  56,  28,   8,  1;
[ 9]  9, 21, 42,  86, 126, 126,  84,  36,  9,  1;
[10] 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1;

Prog

(PARI) T(n, k)=if(n<k|k<0, 0, if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(d, k))))
(SageMath) # uses[DivisorTriangle from A327029]
DivisorTriangle(euler_phi, binomial, 13) # Peter Luschny, Aug 24 2019

Crossrefs

Cf. A053635 (row sums), A121776 (antidiagonal sums), A054630, A327029.

Cf. A000031 (c=2), A001867 (c=3), A001868 (c=4), A001869 (c=5), A054625 (c=6), A054626 (c=7), A054627 (c=8), A054628 (c=9), A054629 (c=10).

Sequence in context: A035517 A099471 A243574 * A127951 A208814 A230449

Adjacent sequences: A121772 A121773 A121774 * A121776 A121777 A121778

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 23 2006

Status

approved