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A255915 Triangle read by rows: T(n,k) = A239672(n)/(A239672(k) * A239672(n-k)). 0

%I

%S 1,1,1,1,63,1,1,728,728,1,1,4032,46592,4032,1,1,15624,999936,999936,

%T 15624,1,1,45864,11374272,62995968,11374272,45864,1,1,117648,85647744,

%U 1838132352,1838132352,85647744,117648,1,1,258048,481886208,30358831104,117640470528

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

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

%C Another name might be the 6-totienomial coefficients.

%H Tom Edgar, <a href="http://www.emis.de/journals/INTEGERS/papers/o62/o62.Abstract.html">Totienomial Coefficients</a>, INTEGERS, 14 (2014), #A62.

%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

%H Donald E. Knuth and Herbert S. Wilf, <a href="http://www.math.upenn.edu/~wilf/website/dm36.pdf">The power of a prime that divides a generalized binomial coefficient</a>, J. Reine Angew. Math., 396:212-219, 1989.

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

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

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

%e 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.

%e The triangle begins:

%e 1;

%e 1, 1;

%e 1, 63, 1;

%e 1, 728, 728, 1;

%e 1, 4032, 46592, 4032, 1;

%e 1, 15624, 999936, 999936, 15624, 1;

%e 1, 45864, 11374272, 62995968, 11374272, 45864, 1

%o (Sage)

%o q=100 #change q for more rows

%o P=[0]+[i^6*prod([1-1/p^6 for p in prime_divisors(i)]) for i in [1..q]]

%o 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.

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

%K nonn,tabl

%O 0,5

%A _Tom Edgar_, Mar 10 2015

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Last modified September 26 11:29 EDT 2020. Contains 337364 sequences. (Running on oeis4.)