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 A000436 Generalized Euler numbers c(3,n). (Formerly M4584 N1955) 15
 1, 8, 352, 38528, 7869952, 2583554048, 1243925143552, 825787662368768, 722906928498737152, 806875574817679474688, 1118389087843083461066752, 1884680130335630169428983808, 3794717805092151129643367268352 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..100 Michael E. Hoffman, Derivative polynomials, Euler Polynomials, and associated integer sequences, El. J. Combinat. 6 (see Th. 3.3). D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694. D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699. D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy] FORMULA E.g.f.: cos(x) / cos(3*x) (even powers only). For n>0, a(n) = A002114(n)*2^(2n+1) = (1/3)*A002112(n)*2^(2n+1). - Philippe Deléham, Jan 17 2004 a(n) = Sum_{k=0..n} (-1)^k*9^(n-k)*A086646(n,k). - Philippe Deléham, Oct 27 2006 (-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 a(n) = P_{2n}(sqrt(3))/sqrt(3) (where the polynomials P_n() are defined in A155100). - N. J. A. Sloane, Nov 05 2009 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 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 a(n) = | 3^(2*n)*2^(2*n+1)*lerchphi(-1,-2*n,1/3) |. - Peter Luschny, Apr 27 2013 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 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 From Vaclav Kotesovec, May 05 2020: (Start) 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)). 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) 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 EXAMPLE G.f. = 1 + 8*x + 352*x^2 + 38528*x^3 + 7869952*x^4 + 2583554048*x^5 + ... MAPLE 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: A000436(10) ; # R. J. Mathar, Nov 19 2006 a := n -> 2*(-144)^n*(Zeta(0, -2*n, 1/6)-Zeta(0, -2*n, 2/3)): seq(a(n), n=0..12); # Peter Luschny, Mar 11 2015 MATHEMATICA 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 *) 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 *) PROG (Sage) from mpmath import mp, lerchphi mp.dps = 32; mp.pretty = True def A000436(n): return abs(3^(2*n)*2^(2*n+1)*lerchphi(-1, -2*n, 1/3)) [A000436(n) for n in (0..12)] # Peter Luschny, Apr 27 2013 (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 CROSSREFS Bisections: A156177 and A156178. Cf. A000191, A007289, overview in A349264. Cf. A210657, A255882. Sequence in context: A158363 A221044 A221163 * A015507 A167256 A277656 Adjacent sequences: A000433 A000434 A000435 * A000437 A000438 A000439 KEYWORD nonn AUTHOR STATUS approved

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Last modified March 28 03:48 EDT 2023. Contains 361577 sequences. (Running on oeis4.)