A006846 Hammersley's polynomial p_n(1).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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)

Crossrefs

Cf. A065547, A085707, A104027, A104030.

Programs

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

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 *)

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

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