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A000436 Generalized Euler numbers c(3,n).
(Formerly M4584 N1955)
13
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) 663-688.

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 Deleham_, Jan 17 2004

a(n) = Sum_{k, 0<=k<=n}(-1)^k*9^(n-k)*A086646(n,k). - _Philippe Deleham_, Oct 27 2006

(-1)^n a(n) = 1-sum_{i=0,1,...,n-1} (-1)^i*binomial(2n,2i)*3^(2n-2i)*a(i). - R. J. Mathar, Nov 19 2006

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

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-Fran├žois 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.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. A210657, A255882.

Sequence in context: A158363 A221044 A221163 * A015507 A167256 A277656

Adjacent sequences:  A000433 A000434 A000435 * A000437 A000438 A000439

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 19 14:06 EDT 2017. Contains 290808 sequences.