A121775 - OEIS

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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

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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