A006846 Hammersley's polynomial p_n(1). (Formerly M1807)

1, 1, 2, 7, 41, 376, 5033, 92821, 2257166, 69981919, 2694447797, 126128146156, 7054258103921, 464584757637001, 35586641825705882, 3136942184333040727, 315295985573234822561, 35843594275585750890976, 4575961401477587844760793, 651880406652100451820206941

Offset

0,3

Comments

Equals column 0 of triangle A104027. Also equals column 0 of triangle A104030 (offset 1). Both A104027 and A104030 involve the trinomial coefficients. - Paul D. Hanna, Mar 06 2005

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)

Formula

a(n) = Sum_{k>=0} (-1)^(n+k)*A065547(n, k) = Sum_{k>=0} A085707(n, k). - Philippe Deléham, Feb 26 2004

E.g.f.: cosh(sqrt(3)*x/2)/cos(x/2) = Sum_{n>=0} a(n)*x^(2n)/(2n)!. - Paul D. Hanna, Feb 27 2005

a(n) = (-1)^n*A104027(n, 0). a(n+1) = (-1)^(n+1)*A104030(n, 0). - Paul D. Hanna, Mar 06 2005

G.f.: 1/(1-x/(1-x/(1-3x/(1-4x/(1-7x/(1-.../(1-ceiling((n+1)^2/4)*x/(1-... (continued fraction). - Paul Barry, Feb 24 2010

a(n) ~ 4*cosh(sqrt(3)*Pi/2) * (2*n)! / Pi^(2*n+1). - Vaclav Kotesovec, Jun 07 2021

Maple

A006846 := proc(n)
    option remember ;
    if n =0 then
        return 1;
    else
        add(binomial(2*n, 2*m)*procname(m)/(-4)^(n-m), m=0..n-1) ;
        (3/4)^n-% ;
    end if
end proc:
seq(A006846(n), n=0..20) ; # R. J. Mathar, Jan 10 2018

Mathematica

h[n_, x_] := Sum[c[k] x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; a[n_] := Sum[(-1)^(n+k)*c[k], {k, 0, n}] /. eq[n] // First; Table[a[n], {n, 0, 15}]   (* Jean-François Alcover, Oct 02 2013, after Philippe Deléham *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^(2*n))); round((2*n)!*polcoeff(cosh(sqrt(3)*X/2)/cos(X/2), 2*n))} # Paul D. Hanna
(Julia)
function A006846list(len::Int)  # Algorithm of L. Seidel (1877)
    R = Array{BigInt}(len)
    A = fill(BigInt(0), len+1); A[1] = 1
    for n in 1:len
        for k in n:-1:2 A[k] += A[k+1] end
        for k in 2:1:n A[k] += A[k-1] end
        R[n] = A[n]
    end
    return R
end
println(A006846list(20)) # Peter Luschny, Jan 02 2018

Crossrefs

Cf. A104027, A104030.

Sequence in context: A101390 A113144 A320415 * A047864 A173916 A163921

Adjacent sequences: A006843 A006844 A006845 * A006847 A006848 A006849

Keyword

nonn

Author

N. J. A. Sloane

Status

approved