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%I #75 Apr 23 2023 07:28:16
%S 0,-1,-1,0,1,0,-3,0,17,0,-155,0,2073,0,-38227,0,929569,0,-28820619,0,
%T 1109652905,0,-51943281731,0,2905151042481,0,-191329672483963,0,
%U 14655626154768697,0,-1291885088448017715,0,129848163681107301953
%N a(n) = 2*n*(2^n - 1)*zeta(1-n) where in the case n=0 the limit is understood, zeta(s) the Riemann zeta function.
%C Also known as the Genocchi numbers, apart from a(0) and a(1) same as A036968.
%C Consider the difference table of a(n), which is a variant of Seidel's Genocchi table A014781:
%C 0 -1 -1 0 1 0 -3 0 17
%C -1 0 1 1 -1 -3 3 17 -17
%C 1 1 0 -2 -2 6 14 -34 -138
%C 0 -1 -2 0 8 8 -48 -104 448
%C -1 -1 2 8 0 -56 -56 552 1160
%C 0 3 6 -8 -56 0 608 608 -8832
%C 3 3 -14 -48 56 608 0 -9440 -9440
%C 0 -17 -34 104 552 -608 -9440 0 198272
%C -17 -17 138 448 -1160 -8832 9440 198272 0
%C a(n) is an autosequence: its inverse binomial transform is the sequence signed (see A181722). The first column (inverse binomial transform) is 0, followed by -A036968. - _Paul Curtz_, Jul 22 2013
%C a(n+1) = p(0) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 1, ..., n+1. - _Michael Somos_, Apr 23 2014
%H Vincenzo Librandi, <a href="/A226158/b226158.txt">Table of n, a(n) for n = 0..550</a>
%H Peter Luschny, <a href="http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html">The Bernoulli Manifesto.</a>
%F E.g.f.: -2*x/(1+exp(-x)).
%F a(2n) = -A000367(n)*A090648(n). - _Paul Curtz_, Jul 22 2013
%F E.g.f.: -2*x/(1+exp(-x))= -2 - 2*T(0), where T(k) = 4*k-1 + x/( 2 - x/( 4*k+1 + x/( 2 - x/T(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 23 2013
%F G.f.: conjecture: -x/Q(0),where Q(k) = 1 - x*(k+1)/(1 + x*(k+1)/(1 - x*(k+1)/(1 + x*(k+1)/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 23 2013
%F a(n) = 2*(1 - 2^n)*Bernoulli(n, 1). - _Peter Luschny_, Apr 16 2014
%F a(n) = -n*Euler(n - 1, 1). - _Michael Somos_, Apr 23 2014
%F a(n) = 2^n*(Bernoulli(n, 1/2) - Bernoulli(n, 1)). - _Peter Luschny_, Jul 10 2020
%F a(n) = 2*n*PolyLog[1 - n, -1] - _Peter Luschny_, Aug 17 2021
%e G.f. = - x - x^2 + x^4 - 3*x^6 + 17*x^8 - 155*x^10 + 2073*x^12 - 38227*x^14 + ...
%p seq(n!*coeff(series(-2*x/(1+exp(-x)), x, 34), x, n), n=0..32);
%p # Second program:
%p A226158 := proc(n) local f; f := z -> Zeta(1-z)*2*z*(2^z-1);
%p if n=0 then limit(f(z), z=0) else f(n) fi end: seq(A226158(n), n=0..32);
%t a[0]=0; a[1]= -1; a[n_]:= n*EulerE[n-1, 0]; Table[a[n], {n,0,32}] (* _Jean-François Alcover_, Sep 12 2013 *)
%t (* Programs from _Michael Somos_, Apr 23 2014 *)
%t a[n_]:= If[n<1, 0, -n*EulerE[n-1, 1]];
%t a[n_]:= If[n<0, 0, 2*(1-2^n)*BernoulliB[n,1]]; (* End *)
%t Table[2*n*PolyLog[1-n, -1], {n,0,32}] (* _Peter Luschny_, Aug 17 2021 *)
%o (Sage)
%o def A226158(n): return -2*n*zeta(1-n)*(1-2^n) if n != 0 else 0
%o [A226158(n) for n in (0..32)]
%o # Alternatively:
%o def A226158_list(len):
%o e, f, R, C = 4, 1, [0], [1]+[0]*(len-1)
%o for n in (2..len-1):
%o for k in range(n, 0, -1):
%o C[k] = -C[k-1] / (k+1)
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append((2-e)*f*C[0])
%o f *= n; e *= 2
%o return R
%o print(A226158_list(34)) # _Peter Luschny_, Feb 22 2016
%o (PARI) my(x='x+O('x^40)); concat([0], Vec(serlaplace(-2*x/(1+exp(-x))))) \\ _G. C. Greubel_, Jan 19 2018
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 50); [0] cat Coefficients(R!(Laplace( -2*x/(1+Exp(-x)) ))); // _G. C. Greubel_, Apr 22 2023
%Y Cf. A000367, A005439, A014781, A036968, A090648, A181722, A227577.
%K sign
%O 0,7
%A _Peter Luschny_, Jun 28 2013
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