A358111 The multiplicative inverse of the coefficients of the factorially normalized Bernoulli polynomials (provided they do not vanish, otherwise by convention 0).

1, -2, 1, 12, -2, 2, 0, 12, -4, 6, -720, 0, 24, -12, 24, 0, -720, 0, 72, -48, 120, 30240, 0, -1440, 0, 288, -240, 720, 0, 30240, 0, -4320, 0, 1440, -1440, 5040, -1209600, 0, 60480, 0, -17280, 0, 8640, -10080, 40320, 0, -1209600, 0, 181440, 0, -86400, 0, 60480, -80640, 362880

Offset: 0

Author

Peter Luschny, Oct 30 2022

Keywords

• sign
• tabl

Comments

The factorially normalized Bernoulli polynomials are defined inductively by:

b(0, x) = 1, (d/dx) b(n, x) = b(n-1, x), and Integral_{x=0..1} b(n, x) = 0.

Examples

			0]        1;
1]       -2,        1;
2]       12,       -2,     2;
3]        0,       12,    -4,      6;
4]     -720,        0,    24,    -12,     24;
5]        0,     -720,     0,     72,    -48,   120;
6]    30240,        0, -1440,      0,    288,  -240,   720;
7]        0,    30240,     0,  -4320,      0,  1440, -1440,   5040;
8] -1209600,        0, 60480,      0, -17280,     0,  8640, -10080, 40320;
9]        0, -1209600,     0, 181440,      0, -86400,    0,  60480,-80640, 362880;
		

References

• N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, 1924. (page 31)

Links

• Yassine El Maazouz and Jim Pitman, The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution, arXiv:2210.02027 [math.PR], Oct. 2022.

Crossrefs

Cf. A227830 (column 0), A196838/A196839 (Bernoulli polynomials), A000142.

Programs

Maple

T := proc(n, k) coeff(bernoulli(n, x) / n!, x, k); ifelse(% = 0, 0, 1/%) end:
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

T[n_, k_] := With[{c = Coefficient[BernoulliB[n, x]/n!, x, k]}, If[c == 0, 0, 1/c]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2024, after Maple code *)

Formula

T(n, k) = ([x^k] b(n, x))^(-1), provided [x^k] b(n, x) != 0, otherwise 0.

Integral_{x=0..1} b(n, x)*b(m, x) = (-1)^(m + 1)*b(n + m, 1). [N. E. Nørlund]

Note that n!*b(n, 1) are the Bernoulli numbers (with B_1 = 1/2).