A321330 -id:A321330 - cp's OEIS frontend

A321329 One third of the numerators of a Boas-Buck sequence for the triangular Sheffer matrix S2[3,1] = A282629.

1, 1, 0, -3, 0, 9, 0, -81, 0, 81, 0, -167913, 0, 2187, 0, -23731137, 0, 287811387, 0, -10310604939, 0, 13761310401, 0, -125613568885131, 0, 3146863577139, 0, -5409187422305481, 0, 8241860346410471769

Offset: 0

Wolfdieter Lang, Nov 15 2018

The denominators are given in A321330.
The general Boas-Buck type recurrence for lower triangular Sheffer matrices S(n, m) is: S(n, m) = (n!/(n-m))*Sum_{k=m..n-1} (1/k!)*(alpha(n-1-k) + m*beta(n-1-k))*S(k, m), for n >= m + 1 >= 1, and inputs S(n, n).
See the Boas-Buck type recurrence for the columns of S2[3,1] = A282629.
For S2[3,1] the Boas-Buck sequence alpha is {1, repeat(0)}.

			The rationals beta begin: {3/2, 3/4, 0, -9/80, 0, 27/1120, 0, -243/44800, 0, 243/197120, 0, -503739/1793792000, 0, 6561/102502400, 0, -71193411/4879114240000, 0, 863434161/259568877568000, 0, -30931814817/40789395046400000, 0, ...}.

Cf. A027641/A027642, A282629, A321330.

a[n_] := Numerator[(-3)^(n+1)*BernoulliB[n+1]/(n+1)!/3]; Array[a, 30, 0] (* Amiram Eldar, Nov 15 2018 *)

a(n) = (1/3)*numerator(beta(n)), with beta(n) = (-3)^{n+1}* B(n+1)/(n+1)!, where B(n) = A027641(n)/A027642(n) (Bernoulli).

G.f. for rationals {beta(n)}_{n >= 0} is d/dx(log((exp(3*x) - 1)/x)) = (3*x*e^(3*x) - e^(3*x) + 1)/(x*(e^(3*x)-1)).

cp's OEIS Frontend

A321329 One third of the numerators of a Boas-Buck sequence for the triangular Sheffer matrix S2[3,1] = A282629.

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