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A000436
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Generalized Euler numbers c(3,n).
(Formerly M4584 N1955)
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15
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1, 8, 352, 38528, 7869952, 2583554048, 1243925143552, 825787662368768, 722906928498737152, 806875574817679474688, 1118389087843083461066752, 1884680130335630169428983808, 3794717805092151129643367268352
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OFFSET
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0,2
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REFERENCES
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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).
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LINKS
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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]
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FORMULA
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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
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EXAMPLE
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G.f. = 1 + 8*x + 352*x^2 + 38528*x^3 + 7869952*x^4 + 2583554048*x^5 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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