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

%I A358111 #13 Mar 07 2024 08:31:58
%S A358111 1,-2,1,12,-2,2,0,12,-4,6,-720,0,24,-12,24,0,-720,0,72,-48,120,30240,
%T A358111 0,-1440,0,288,-240,720,0,30240,0,-4320,0,1440,-1440,5040,-1209600,0,
%U A358111 60480,0,-17280,0,8640,-10080,40320,0,-1209600,0,181440,0,-86400,0,60480,-80640,362880
%N A358111 The multiplicative inverse of the coefficients of the factorially normalized Bernoulli polynomials (provided they do not vanish, otherwise by convention 0).
%C A358111 The factorially normalized Bernoulli polynomials are defined inductively by:
%C A358111 b(0, x) = 1, (d/dx) b(n, x) = b(n-1, x), and Integral_{x=0..1} b(n, x) = 0.
%D A358111 N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, 1924. (page 31)
%H A358111 Yassine El Maazouz and Jim Pitman, <a href="https://doi.org/10.48550/arxiv.2210.02027">The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution</a>, arXiv:2210.02027 [math.PR], Oct. 2022.
%F A358111 T(n, k) = ([x^k] b(n, x))^(-1), provided [x^k] b(n, x) != 0, otherwise 0.
%F A358111 Integral_{x=0..1} b(n, x)*b(m, x) = (-1)^(m + 1)*b(n + m, 1). [N. E. Nørlund]
%F A358111 Note that n!*b(n, 1) are the Bernoulli numbers (with B_1 = 1/2).
%e A358111 0]        1;
%e A358111 1]       -2,        1;
%e A358111 2]       12,       -2,     2;
%e A358111 3]        0,       12,    -4,      6;
%e A358111 4]     -720,        0,    24,    -12,     24;
%e A358111 5]        0,     -720,     0,     72,    -48,   120;
%e A358111 6]    30240,        0, -1440,      0,    288,  -240,   720;
%e A358111 7]        0,    30240,     0,  -4320,      0,  1440, -1440,   5040;
%e A358111 8] -1209600,        0, 60480,      0, -17280,     0,  8640, -10080, 40320;
%e A358111 9]        0, -1209600,     0, 181440,      0, -86400,    0,  60480,-80640, 362880;
%p A358111 T := proc(n, k) coeff(bernoulli(n, x) / n!, x, k); ifelse(% = 0, 0, 1/%) end:
%p A358111 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A358111 T[n_, k_] := With[{c = Coefficient[BernoulliB[n, x]/n!, x, k]}, If[c == 0, 0, 1/c]];
%t A358111 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 07 2024, after Maple code *)
%Y A358111 Cf. A227830 (column 0), A196838/A196839 (Bernoulli polynomials), A000142.
%K A358111 sign,tabl
%O A358111 0,2
%A A358111 _Peter Luschny_, Oct 30 2022