|
| |
|
|
A000436
|
|
Generalized Euler numbers.
(Formerly M4584 N1955)
|
|
9
| |
|
|
1, 8, 352, 38528, 7869952, 2583554048, 1243925143552, 825787662368768, 722906928498737152, 806875574817679474688, 1118389087843083461066752, 1884680130335630169428983808, 3794717805092151129643367268352
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
REFERENCES
| D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.
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
| Michael E. Hoffman, DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES (see Th. 3.3)
|
|
|
FORMULA
| E.g.f.: cos x / cos 3x.
For n>0, a(n) = A002114(n)*2^(2n+1) = (1/3)*A002112(n)*2^(2n+1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 17 2004
a(n)=Sum_{k, 0<=k<=n}(-1)^k*9^(n-k)*A086646(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 (mathar(AT)strw.leidenuniv.nl), 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
|
|
|
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 (mathar(AT)strw.leidenuniv.nl), Nov 19 2006
|
|
|
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}] (* From Jean-François Alcover, Jan 31 2012, after R. J. Mathar *)
|
|
|
CROSSREFS
| Bisections: A156177 and A156178.
Sequence in context: A079485 A193504 A158363 * A015507 A167256 A193837
Adjacent sequences: A000433 A000434 A000435 * A000437 A000438 A000439
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
