A273192 -id:A273192 - cp's OEIS frontend

A273194 a(n) = numerator(R(n,3)), where R(n,d) = (Product_{j prime to d} Pochhammer(j/d, n)) / n!.

1, 2, 20, 1120, 30800, 1121120, 152472320, 8277068800, 523524601600, 340290991040000, 27631628472448000, 2491870494969856000, 741331472253532160000, 80177849999112785920000, 9392262428467497779200000, 3554032102932101159649280000, 480238587908700169197608960000

Offset: 0

Peter Luschny, Jun 06 2016

Also the numerators of the nonzero coefficients in the expansion of hypergeom([Seq_{k=1..m-1} k/m], [], (x/m)^m) for m = 3.

R(n, 1) = A000012 / A000012.
R(n, 2) = A001790 / A046161.
R(n, 3) = (this sequence) / A344402.
Cf. A273192, A273193.

Hlist := proc(m,size) local H, S;
H := m -> hypergeom([seq(k/m, k=1..m-1)], [], (x/m)^m);
S := m -> series(H(m), x, (m+1)*size);
seq(numer(coeff(S(m), x, m*n)), n=0..size) end:
A273194_list := size -> Hlist(3, size);
# Alternative:
coprimes := n -> select(j -> igcd(j, n) = 1, {$1..n}):
R := (n, d) -> mul(pochhammer(j/d, n), j in coprimes(d)) / n!:
seq(numer(R(n, 3)), n = 0..16); # Peter Luschny, May 20 2021

A273193 Numerators of the nonzero coefficients in the expansion of 1/hypergeom([Seq_{k=1..m-1} k/m], [], (x/m)^m) for m = 3.

Original entry on oeis.org

1, 2, -16, 904, -25792, 971936, -135875584, 7531512832, -483853915136, 318210896625152, -26070230641872896, 2367374418301892608, -708155254090757373952, 76928188353501090512896, -9044296958948501037252608, 3432739126498593173574909952, -465041552940366454298747600896

Offset: 0

Peter Luschny, Jun 06 2016

Cf. A000012 (m=1), A002596 (m=2), A273192, A273194.

Blist := proc(m, size) local H, S;
H := m -> hypergeom([seq(k/m, k=1..m-1)], [], (x/m)^m);
S := m -> series(1/H(m), x, (m+1)*size);
seq((-1)^n*numer(coeff(S(m), x, m*n)), n=0..size) end:
A273193_list := size -> Blist(3, size);

cp's OEIS Frontend

A273194 a(n) = numerator(R(n,3)), where R(n,d) = (Product_{j prime to d} Pochhammer(j/d, n)) / n!.

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