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A000436 Generalized Euler numbers c(3,n).

%I A000436 M4584 N1955 #92 Oct 26 2024 22:58:57
%S A000436 1,8,352,38528,7869952,2583554048,1243925143552,825787662368768,
%T A000436 722906928498737152,806875574817679474688,1118389087843083461066752,
%U A000436 1884680130335630169428983808,3794717805092151129643367268352
%N A000436 Generalized Euler numbers c(3,n).
%D A000436 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000436 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000436 T. D. Noe, <a href="/A000436/b000436.txt">Table of n, a(n) for n = 0..100</a>
%H A000436 Michael E. Hoffman, <a href="https://doi.org/10.37236/1453">Derivative polynomials, Euler Polynomials, and associated integer sequences</a>, El. J. Combinat. 6 (see Th. 3.3).
%H A000436 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967) 689-694.
%H A000436 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0227093-9">Corrigenda to: "Generalized Euler and class numbers"</a>, Math. Comp. 22 (1968), 699.
%H A000436 D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%F A000436 E.g.f.: cos(x) / cos(3*x) (even powers only).
%F A000436 For n>0, a(n) = A002114(n)*2^(2n+1) = (1/3)*A002112(n)*2^(2n+1). - _Philippe Deléham_, Jan 17 2004
%F A000436 a(n) = Sum_{k=0..n} (-1)^k*9^(n-k)*A086646(n,k). - _Philippe Deléham_, Oct 27 2006
%F A000436 (-1)^n a(n) = 1 - Sum_{i=0..n-1} (-1)^i*binomial(2n,2i)*3^(2n-2i)*a(i). - _R. J. Mathar_, Nov 19 2006
%F A000436 a(n) = P_{2n}(sqrt(3))/sqrt(3) (where the polynomials P_n() are defined in A155100). - _N. J. A. Sloane_, Nov 05 2009
%F A000436 E.g.f.: E(x) = cos(x)/cos(3*x) = 1 + 4*x^2/(G(0)-2*x^2); G(k) = (2*k+1)*(k+1) - 2*x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction, Euler's kind, 1-step). - _Sergei N. Gladkovskii_, Jan 02 2012
%F A000436 G.f.: 1 / (1 - 2*4*x / (1 - 6*6*x / (1 - 8*10*x / (1 - 12*12*x / (1 - 14*16*x / (1 - 18*18*x / ...)))))). - _Michael Somos_, May 12 2012
%F A000436 a(n) = | 3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3) |. - _Peter Luschny_, Apr 27 2013
%F A000436 a(n) = (-1)^n*6^(2*n)*E(2*n,1/3), where E(n,x) denotes the n-th Euler polynomial. Calculation suggests that the expansion exp( Sum_{n >= 1} a(n)*x^n/n ) = exp( 8*x + 352*x^2/2 + 38528*x^3/3 + ... ) = 1 + 8*x + 208*x^2 + 14336*x^3 + ... has integer coefficients. Cf. A255882. - _Peter Bala_, Mar 10 2015
%F A000436 a(n) = 2*(-144)^n*(zeta(-2*n,1/6)-zeta(-2*n,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - _Peter Luschny_, Mar 11 2015
%F A000436 From _Vaclav Kotesovec_, May 05 2020: (Start)
%F A000436 For n>0, a(n) = (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (sqrt(3)*Pi^(2*n+1)).
%F A000436 For n>0, a(n) = (-1)^(n+1) * 2^(2*n-1) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (Pi*sqrt(3)*zeta(2*n)). (End)
%F A000436 Conjecture: for each positive integer k, the sequence defined by a(n) (mod k) is eventually periodic with period dividing phi(k). For example, modulo 13 the sequence becomes [1, 8, 1, 9, 12, 10, 0, 8, 1, 9, 12, 10, 0, ...]; after the initial term 1 this appears to be a periodic sequence of period 6, a divisor of phi(13) = 12. - _Peter Bala_, Dec 11 2021
%e A000436 G.f. = 1 + 8*x + 352*x^2 + 38528*x^3 + 7869952*x^4 + 2583554048*x^5 + ...
%p A000436 A000436 := proc(nmax) local a,n,an; a := [1] : n := 1 : while nops(a)< nmax do an := 1-sum(binomial(2*n,2*i)*3^(2*n-2*i)*(-1)^i*op(i+1,a),i=0..n-1) : a := [op(a),an*(-1)^n] ; n := n+1 ; od ; RETURN(a) ; end:
%p A000436 A000436(10) ; # _R. J. Mathar_, Nov 19 2006
%p A000436 a := n -> 2*(-144)^n*(Zeta(0,-2*n,1/6)-Zeta(0,-2*n,2/3)):
%p A000436 seq(a(n), n=0..12); # _Peter Luschny_, Mar 11 2015
%t A000436 a[0] = 1; a[n_] := a[n] = (-1)^n*(1 - Sum[(-1)^i*Binomial[2n, 2i]*3^(2n - 2i)*a[i], {i, 0, n-1}]); Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Jan 31 2012, after _R. J. Mathar_ *)
%t A000436 With[{nn=30},Take[CoefficientList[Series[Cos[x]/Cos[3x],{x,0,nn}], x] Range[ 0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, May 22 2012 *)
%o A000436 (Sage)
%o A000436 from mpmath import mp, lerchphi
%o A000436 mp.dps = 32; mp.pretty = True
%o A000436 def A000436(n): return abs(3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3))
%o A000436 [A000436(n) for n in (0..12)]  # _Peter Luschny_, Apr 27 2013
%o A000436 (PARI) x='x+O('x^66); v=Vec(serlaplace( cos(x) / cos(3*x) ) ); vector(#v\2,n,v[2*n-1]) \\ _Joerg Arndt_, Apr 27 2013
%Y A000436 Row 3 of A235605.
%Y A000436 Bisections: A156177 and A156178.
%Y A000436 Cf. A000191, A007289, overview in A349264.
%Y A000436 Cf. A210657, A255882.
%K A000436 nonn
%O A000436 0,2
%A A000436 _N. J. A. Sloane_