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

0,5

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.

Links

Table of n, a(n) for n=0..40.

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.

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)).

Example

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

Prog

(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.

Crossrefs

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

Sequence in context: A116270 A324018 A102241 * A351341 A234692 A339458

Adjacent sequences: A255912 A255913 A255914 * A255916 A255917 A255918

Keyword

nonn,tabl

Author

Tom Edgar, Mar 10 2015

Status

approved