A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

1, -3, 3, 0, -27, 0, 243, 0, -2187, 0, 98415, 0, -122408577, 0, 11160261, 0, -51899996619, 0, 5664991530321, 0, -202943637014337, 0, 8938507796555139, 0, -22252066887294301257, 0, 7246946747292751629, 0, -181103830292539169071623

Offset: 0

Author

Wolfdieter Lang, Apr 29 2017

Keywords

• sign
• easy
• frac

Comments

The denominators are given in A285068.

In general the numbers B(d;n) = d^n*B(n), for n >= 0, have e.g.f. d*x/(exp(d*x) - 1). They are also the exponential convolution of the generalized Bernoulli numbers B[d,a](n), obtained from the generalized Stirling2 numbers S2[d,a], with the sequence {(-a)^n}_{n>=0}. See a comment in A157817 for the B[4,1] and B[4,3] examples.

These numbers B(d;n) and their polynomials B(d;n,x) = Sum_{m=0..n} binomial(n, m)*B(d;n-m)*x^m are used in the generalized so-called Faulhaber formula for the sums of powers of arithmetic progressions defined by SP(d,a;n,m) := Sum_{j=0..m} (a + d*j)^n = Sum_{k=0..n} binomial(n, k)*a^(n-k)*d^k*SP(k,m) with SP(k,m) = SP(1,0;k,m), n >= 0, m >= 0, and 0^0 := 1.

The Faulhaber formula is: SP(d,a;n,m) = (1/(d*(n+1)))*[B(d;n+1,x = a+d*(m+1)) - B(d;n+1,x = d) - B(d;n+1,x = a) + B(d;n+1,x=0) + d^(n+1)*[n=0]]. Here [n=0] is the Kronecker delta_{n,0} symbol: 1 if n=0 and 0 otherwise.

A simpler version of the Faulhaber formula is for a=0: SP(d,0;0,m) = m+1 and SP(d,0;n,m) = d^n*(1/(n+1))*(B(n+1, x = m+1) - B(n+1, x=1)) for n >= 1, and for a an integer >= 1: Sum_{k=0..n} binomial(n, k)*a^(n-k) * d^k * (1/(k+1)) * (B(k+1, x=m+1) - B(k+1, x=1)). Here B(n, x) = B(1;n,x) are the usual Bernoulli polynomials from A196838/A196839 or A053382/A053383.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..300
• Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Crossrefs

Cf. A027641/A027642, A144845, A285068.

Programs

Maple

seq(numer(3^n*bernoulli(n)), n=0..28); # Peter Luschny, Jul 17 2017

Mathematica

Table[Numerator[3^n*BernoulliB[n]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

PARI

a(n) = numerator(3^n * bernfrac(n)); \\ Ruud H.G. van Tol, Jan 31 2024

Python

from sympy import bernoulli
def a(n): return -3 if n == 1 else (3**n * bernoulli(n)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

Formula

a(n) = numerator(r(n)) with r(n) = 3^n*A027641(n)/A027642(n), n >= 0.

E.g.f. {r(n)}_{n>=0}: 3*x/(exp(3*x) - 1).