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 A238498 Triangle read by rows: T(n,k) = A175836(n)/(A175836(k)* A175836(n-k)). 4
 1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 8, 6, 1, 1, 6, 12, 12, 6, 1, 1, 12, 24, 36, 24, 12, 1, 1, 8, 32, 48, 48, 32, 8, 1, 1, 12, 32, 96, 96, 96, 32, 12, 1, 1, 12, 48, 96, 192, 192, 96, 48, 12, 1, 1, 18, 72, 216, 288, 576, 288, 216, 72, 18, 1, 1, 12, 72, 216, 432, 576, 576, 432, 216, 72, 12, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS We assume that A175836(0)=1 since it would be the empty product. These are the generalized binomial coefficients associated with the Dedekind psi function A001615. Another name might be the psi-nomial coefficients. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened 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) = A175836(n)/(A175836(k)*A175836(n-k)). T(n,k) = prod_{i=1..n} A001615(i)/(prod_{i=1..k} A001615(i)*prod_{i=1..n-k} A001615(i)). T(n,k) = A001615(n)/n*(k/A001615(k)*T(n-1,k-1)+(n-k)/A001615(n-k)*T(n-1,k)). T(n,k) = A238688(n,k)/A238453(n,k). EXAMPLE The first five terms in the Dedekind psi function are 1,3,4,6,6 and so T(4,2) = 6*4*3*1/((3*1)*(3*1))=8 and T(5,3) = 6*6*4*3*1/((4*3*1)*(3*1))=12. The triangle begins 1 1  1 1  3  1 1  4  4  1 1  6  8  6  1 1  6  12 12 6 1 MAPLE A175836 := proc(n) option remember; local p; `if`(n<2, 1, n*mul(1+1/p, p=factorset(n))*A175836(n-1)) end: A238498 := (n, k) -> A175836(n)/(A175836(k)*A175836(n-k)): seq(seq(A238498(n, k), k=0..n), n=0..10); # Peter Luschny, Feb 28 2014 MATHEMATICA DedekindPsi[n_] := Sum[MoebiusMu[n/d] d^2 , {d, Divisors[n]}]/EulerPhi[n]; (* b = A175836 *) b[n_] := Times @@ DedekindPsi /@ Range[n]; T[n_, k_] := b[n]/(b[k] b[n-k]); Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jul 02 2019 *) PROG (Sage) q=100 #change q for more rows P=[0]+[i*prod([(1+1/x) for x 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. (Haskell) a238498 n k = a238498_tabl !! n !! k a238498_row n = a238498_tabl !! n a238498_tabl = [1] : f [1] a001615_list where    f xs (z:zs) = (map (div y) \$ zipWith (*) ys \$ reverse ys) : f ys zs      where ys = y : xs; y = head xs * z -- Reinhard Zumkeller, Mar 01 2014 CROSSREFS Cf. A001615, A175836, A238453. Sequence in context: A147290 A026670 A131402 * A026626 A136482 A026648 Adjacent sequences:  A238495 A238496 A238497 * A238499 A238500 A238501 KEYWORD nonn,tabl AUTHOR Tom Edgar, Feb 27 2014 STATUS approved

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Last modified January 23 17:33 EST 2022. Contains 350514 sequences. (Running on oeis4.)