A255915 Triangle read by rows: T(n,k) = A239672(n)/(A239672(k) * A239672(n-k)).

1, 1, 1, 1, 63, 1, 1, 728, 728, 1, 1, 4032, 46592, 4032, 1, 1, 15624, 999936, 999936, 15624, 1, 1, 45864, 11374272, 62995968, 11374272, 45864, 1, 1, 117648, 85647744, 1838132352, 1838132352, 85647744, 117648, 1, 1, 258048, 481886208, 30358831104, 117640470528

Offset: 0

Author

Tom Edgar, Mar 10 2015

Keywords

• nonn
• tabl

Comments

These are the generalized binomial coefficients associated with the Jordan totient function J_6 given in A069091.

Another name might be the 6-totienomial coefficients.

Examples

			The first five terms in the sixth Jordan totient function are 1, 63, 728, 4032, 15624 and so T(4,2) = 4032*728*63*1/((63*1)*(63*1)) = 46592 and T(5,3) = 15624*4032*728*63*1/((728*63*1)*(63*1)) = 999936.
The triangle begins:
1;
1, 1;
1, 63, 1;
1, 728, 728, 1;
1, 4032, 46592, 4032, 1;
1, 15624, 999936, 999936, 15624, 1;
1, 45864, 11374272, 62995968, 11374272, 45864, 1
		

Links

• Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
• Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
• Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Crossrefs

Cf. A069091, A239672, A238453, A238688, A238743, A238754, A239633.

Programs

Sage

q=100 #change q for more rows
P=[0]+[i^6*prod([1-1/p^6 for p in prime_divisors(i)]) for i in [1..q]]
Triangle=[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

Formula

T(n,k) = A239672(n)/(A239672(k) * A239672(n-k)).

T(n,k) = Product_{i=1..n} A069091(i)/(Product_{i=1..k} A069091(i)*Product_{i=1..n-k} A069091(i)).

T(n,k) = A069091(n)/n*(k/A069091(k)*T(n-1,k-1)+(n-k)/A069091(n-k)*T(n-1,k)).