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

1, 0, -1, 0, 3, 0, -5, 0, 7, 0, -45, 0, 7601, 0, -91, 0, 54255, 0, -745739, 0, 3317609, 0, -17944773, 0, 5436374093, 0, -213827575, 0, 641235447783, 0, -249859397004145, 0, 238988952277727, 0, -85063699326111, 0, 921034504356871708055, 0, -108409774812137683

Offset: 0

Author

Peter Luschny, Apr 30 2009

Keywords

• sign

Comments

Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function

(t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!

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

N1_n = A027641, D1_n = A141056 (Clausen).

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

N2_n = this sequence, D2_n = A141056 (Clausen).

Examples

			The Clausen-normalized Bernoulli polynomials of order 2 are:
1
2 x - 2
6 x^2 - 12 x + 5
2 x^3 - 6 x^2 + 5 x - 1
30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
42 x^6 - 252 x^5 + 525 x^4 - 420 x^3 + 63 x^2 + 42 x - 5
The value of these polynomials at x = 1 gives the sequence.
		

References

• L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
• C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
• N. E. Nørlund, Vorlesungen über Differenzenrechnung, Berlin, Springer-Verlag, 1924.

Crossrefs

Cf. A120282, A132094, A100615 and A027643.

Programs

Maple

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);

Mathematica

aList[n_] := Module[{g, c, s},
   g[k_] := (t/(Exp[t]-1))^k*Exp[x*t];
   c[k_] := Times @@ Select[Divisors[k]+1, PrimeQ];
   s = Series[g[2], {t, 0, n + 8}] // Normal;
Join[{1}, Table[i!*c[i]*Coefficient[s, t, i] /. x -> 1, {i, 1, n}]]];
aList[38] (* Jean-François Alcover, May 28 2024, after Peter Luschny *)