A239672 -id:A239672 - cp's OEIS frontend

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

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)).

A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 72, 432, 5184, 41472, 497664, 5971968, 107495424, 1289945088, 30958682112, 433421549568, 10402117189632, 249650812551168, 5991619501228032, 107849151022104576, 3882569436795764736

Offset: 1

Enrique Pérez Herrero, Sep 18 2010

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A060648(gcd(i,j)) for 1 <= i,j <= n, note that A060648 is the Inverse Möbius transform of A001615. - Enrique Pérez Herrero, Aug 12 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..423
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
Eric Weisstein's World of Mathematics, Le Paige's Theorem
Index to divisibility sequences

Cf. A001615, A001088, A059381, A059382, A059383, A059384, A238498.

Cf. A239672.

a175836 n = a175836_list !! (n-1)
a175836_list = scanl1 (*) a001615_list
-- Reinhard Zumkeller, Mar 01 2014

A175836 := proc(n) option remember; local p; `if`(n<2,1, n*mul(1+1/p,p=factorset(n))*A175836(n-1)) end: # Peter Luschny, Feb 28 2014

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];
A175836[n_]:=Times@@DedekindPsi/@Range[n]

a=direuler(p=2, 100, (1+X)/(1-p*X));for(i=2,#a,a[i]*=a[i-1]);a
\\ Charles R Greathouse IV, Jul 29 2011

a(n) = A059381(n)/A001088(n).

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

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A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

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