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Hammersley's polynomial p_n(1).
(Formerly M1807)
13

%I M1807 #45 May 02 2024 15:04:27

%S 1,1,2,7,41,376,5033,92821,2257166,69981919,2694447797,126128146156,

%T 7054258103921,464584757637001,35586641825705882,3136942184333040727,

%U 315295985573234822561,35843594275585750890976,4575961401477587844760793,651880406652100451820206941

%N Hammersley's polynomial p_n(1).

%C 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

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

%H Vincenzo Librandi, <a href="/A006846/b006846.txt">Table of n, a(n) for n = 0..100</a>

%H J. M. Hammersley, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/14_1_1.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23.

%H J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)

%F a(n) = Sum_{k>=0} (-1)^(n+k)*A065547(n, k) = Sum_{k>=0} A085707(n, k). - _Philippe Deléham_, Feb 26 2004

%F 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

%F a(n) = (-1)^n*A104027(n, 0). a(n+1) = (-1)^(n+1)*A104030(n, 0). - _Paul D. Hanna_, Mar 06 2005

%F 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

%F a(n) ~ 4*cosh(sqrt(3)*Pi/2) * (2*n)! / Pi^(2*n+1). - _Vaclav Kotesovec_, Jun 07 2021

%p A006846 := proc(n)

%p option remember ;

%p if n =0 then

%p return 1;

%p else

%p add(binomial(2*n,2*m)*procname(m)/(-4)^(n-m),m=0..n-1) ;

%p (3/4)^n-% ;

%p end if

%p end proc:

%p seq(A006846(n),n=0..20) ; # _R. J. Mathar_, Jan 10 2018

%t 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_ *)

%o (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_

%o (Julia)

%o function A006846list(len::Int) # Algorithm of L. Seidel (1877)

%o R = Array{BigInt}(len)

%o A = fill(BigInt(0), len+1); A[1] = 1

%o for n in 1:len

%o for k in n:-1:2 A[k] += A[k+1] end

%o for k in 2:1:n A[k] += A[k-1] end

%o R[n] = A[n]

%o end

%o return R

%o end

%o println(A006846list(20)) # _Peter Luschny_, Jan 02 2018

%Y Cf. A065547, A085707, A104027, A104030.

%K nonn

%O 0,3

%A _N. J. A. Sloane_