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A160035 Clausen-normalized numerators of the Bernoulli numbers of order 2.

%I A160035 #12 May 28 2024 12:36:32
%S A160035 1,0,-1,0,3,0,-5,0,7,0,-45,0,7601,0,-91,0,54255,0,-745739,0,3317609,0,
%T A160035 -17944773,0,5436374093,0,-213827575,0,641235447783,0,
%U A160035 -249859397004145,0,238988952277727,0,-85063699326111,0,921034504356871708055,0,-108409774812137683
%N A160035 Clausen-normalized numerators of the Bernoulli numbers of order 2.
%C A160035 Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function
%C A160035 (t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!
%C A160035 Bernoulli numbers of order 1 (defined as B_n{^(1)}(1)) can be regarded as a pair of sequences B1_n = N1_n / D1_n with
%C A160035 N1_n = A027641, D1_n = A141056 (Clausen).
%C A160035 Similarly Bernoulli numbers of order 2 (defined as B_n{^(2)}(1)) can be regarded as a pair of sequences B2_n = N2_n / D2_n with
%C A160035 N2_n = this sequence, D2_n = A141056 (Clausen).
%D A160035 L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
%D A160035 C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
%D A160035 N. E. Nørlund, Vorlesungen über Differenzenrechnung, Berlin, Springer-Verlag, 1924.
%e A160035 The Clausen-normalized Bernoulli polynomials of order 2 are:
%e A160035 1
%e A160035 2 x - 2
%e A160035 6 x^2 - 12 x + 5
%e A160035 2 x^3 - 6 x^2 + 5 x - 1
%e A160035 30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
%e A160035 2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
%e A160035 42 x^6 - 252 x^5 + 525 x^4 - 420 x^3 + 63 x^2 + 42 x - 5
%e A160035 The value of these polynomials at x = 1 gives the sequence.
%p A160035 aList := proc(n) local g,c,i; g := k -> (t/(exp(t)-1))^k*exp(x*t): c := proc(n) local i; mul(i,i=select(isprime, map(i->i+1,numtheory[divisors](n)))) end: convert(series(g(2),t,n+8),polynom): seq(i!*c(i)*subs(x=1,coeff(%,t,i)),i=0..n) end: aList(38);
%t A160035 aList[n_] := Module[{g, c, s},
%t A160035    g[k_] := (t/(Exp[t]-1))^k*Exp[x*t];
%t A160035    c[k_] := Times @@ Select[Divisors[k]+1, PrimeQ];
%t A160035    s = Series[g[2], {t, 0, n + 8}] // Normal;
%t A160035 Join[{1}, Table[i!*c[i]*Coefficient[s, t, i] /. x -> 1, {i, 1, n}]]];
%t A160035 aList[38] (* _Jean-François Alcover_, May 28 2024, after _Peter Luschny_ *)
%Y A160035 Cf. A120282, A132094, A100615 and A027643.
%K A160035 sign
%O A160035 0,5
%A A160035 _Peter Luschny_, Apr 30 2009