A238743 -id:A238743 - cp's OEIS frontend

A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

1, 1, 1, 1, 15, 1, 1, 80, 80, 1, 1, 240, 1280, 240, 1, 1, 624, 9984, 9984, 624, 1, 1, 1200, 49920, 149760, 49920, 1200, 1, 1, 2400, 192000, 1497600, 1497600, 192000, 2400, 1, 1, 3840, 614400, 9216000, 23961600, 9216000, 614400, 3840, 1, 1, 6480, 1658880

Offset: 0

Tom Edgar, Mar 04 2014

We assume that A059383(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_4 given in A059377.
Another name might be the 4-totienomial coefficients.

			The first five terms in the fourth Jordan totient function are 1,15,80,240,624 and so T(4,2) = 240*80*15*1/((15*1)*(15*1))=1280 and T(5,3) = 624*240*80*15*1/((80*15*1)*(15*1))=9984.
The triangle begins
1
1 1
1 15  1
1 80  80   1
1 240 1280 240  1
1 624 9984 9984 624 1

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.

Cf. A059377, A059383, A238453, A238688, A238743.

q=100 #change q for more rows
P=[0]+[i^4*prod([1-1/p^4 for p in prime_divisors(i)]) for i in [1..q]]
[[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.

T(n,k) = A059383(n)/(A059383(k)* A059383(n-k)).
T(n,k) = prod_{i=1..n} A059377(i)/(prod_{i=1..k} A059377(i)*prod_{i=1..n-k} A059377(i)).
T(n,k) = A059377(n)/n*(k/A059377(k)*T(n-1,k-1)+(n-k)/A059377(n-k)*T(n-1,k)).

A239633 Triangle read by rows: T(n,k) = A059384(n)/(A059384(k)*A059384(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 31, 1, 1, 242, 242, 1, 1, 992, 7744, 992, 1, 1, 3124, 99968, 99968, 3124, 1, 1, 7502, 756008, 3099008, 756008, 7502, 1, 1, 16806, 4067052, 52501944, 52501944, 4067052, 16806, 1, 1, 31744, 17209344, 533489664, 1680062208, 533489664, 17209344, 31744

Offset: 0

Tom Edgar, Mar 22 2014

We assume that A059384(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_5 given in A059378.
Another name might be the 5-totienomial coefficients.

			The first five terms in the fifth Jordan totient function are 1,31,242,992,3124 and so T(4,2) = 992*242*31*1/((31*1)*(31*1))=7744 and T(5,3) = 3124*992*242*31*1/((242*31*1)*(31*1))=99968.
The triangle begins
1
1 1
1 31   1
1 242  242   1
1 992  7744  992   1
1 3124 99968 99968 3124 1

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.

Cf. A059378, A059384, A238453, A238688, A238743, A238754.

q=100 #change q for more rows
P=[0]+[i^5*prod([1-1/p^5 for p in prime_divisors(i)]) for i in [1..q]]
[[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.

T(n,k) = A059384(n)/(A059384(k)* A059384(n-k)).
T(n,k) = prod_{i=1..n} A059378(i)/(prod_{i=1..k} A059378(i)*prod_{i=1..n-k} A059378(i)).
T(n,k) = A059378(n)/n*(k/A059378(k)*T(n-1,k-1)+(n-k)/A059378(n-k)*T(n-1,k)).

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

Original entry on oeis.org

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

Tom Edgar, Mar 10 2015

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.

			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

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.

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

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.

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)).

cp's OEIS Frontend

A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

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A239633 Triangle read by rows: T(n,k) = A059384(n)/(A059384(k)*A059384(n-k)).

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

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