cp's OEIS Frontend

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.

%I A121775 #16 Jul 22 2024 23:30:50
%S A121775 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,
%T A121775 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,
%U A121775 10,27,59,130,215,253,210,120,45,10,1,11,21,55,165,330,462,462,330,165,55
%N 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.
%C A121775 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.
%e A121775 Triangle begins:
%e A121775 [ 0]  1;
%e A121775 [ 1]  1,  1;
%e A121775 [ 2]  2,  3,  1;
%e A121775 [ 3]  3,  5,  3,   1;
%e A121775 [ 4]  4,  8,  7,   4,   1;
%e A121775 [ 5]  5,  9, 10,  10,   5,   1;
%e A121775 [ 6]  6, 15, 20,  21,  15,   6,   1;
%e A121775 [ 7]  7, 13, 21,  35,  35,  21,   7,   1;
%e A121775 [ 8]  8, 20, 36,  60,  71,  56,  28,   8,  1;
%e A121775 [ 9]  9, 21, 42,  86, 126, 126,  84,  36,  9,  1;
%e A121775 [10] 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1;
%o A121775 (PARI) T(n,k)=if(n<k||k<0,0,if(n==0,1,sumdiv(n,d,eulerphi(n/d)*binomial(d,k))))
%o A121775 (SageMath) # uses[DivisorTriangle from A327029]
%o A121775 DivisorTriangle(euler_phi, binomial, 13) # _Peter Luschny_, Aug 24 2019
%Y A121775 Cf. A053635 (row sums), A121776 (antidiagonal sums), A054630, A327029.
%Y A121775 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).
%K A121775 nonn,tabl
%O A121775 0,4
%A A121775 _Paul D. Hanna_, Aug 23 2006