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

%I A285863 #48 Jun 26 2025 15:21:05
%S A285863 1,-3,3,0,-27,0,243,0,-2187,0,98415,0,-122408577,0,11160261,0,
%T A285863 -51899996619,0,5664991530321,0,-202943637014337,0,8938507796555139,0,
%U A285863 -22252066887294301257,0,7246946747292751629,0,-181103830292539169071623
%N A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).
%C A285863 The denominators are given in A285068.
%C A285863 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.
%C A285863 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.
%C A285863 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.
%C A285863 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.
%H A285863 Indranil Ghosh, <a href="/A285863/b285863.txt">Table of n, a(n) for n = 0..300</a>
%H A285863 Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%F A285863 a(n) = numerator(r(n)) with r(n) = 3^n*A027641(n)/A027642(n), n >= 0.
%F A285863 E.g.f. {r(n)}_{n>=0}: 3*x/(exp(3*x) - 1).
%p A285863 seq(numer(3^n*bernoulli(n)), n=0..28); # _Peter Luschny_, Jul 17 2017
%t A285863 Table[Numerator[3^n*BernoulliB[n]], {n, 0, 100}] (* _Indranil Ghosh_, Jul 18 2017 *)
%o A285863 (Python)
%o A285863 from sympy import bernoulli
%o A285863 def a(n): return -3 if n == 1 else (3**n * bernoulli(n)).numerator
%o A285863 print([a(n) for n in range(51)]) # _Indranil Ghosh_, Jul 18 2017
%o A285863 (PARI) a(n) = numerator(3^n * bernfrac(n)); \\ _Ruud H.G. van Tol_, Jan 31 2024
%Y A285863 Cf. A027641/A027642, A144845, A285068.
%K A285863 sign,easy,frac
%O A285863 0,2
%A A285863 _Wolfdieter Lang_, Apr 29 2017