cp's OEIS Frontend

A006846 Hammersley's polynomial p_n(1).

%I A006846 M1807 #45 May 02 2024 15:04:27
%S A006846 1,1,2,7,41,376,5033,92821,2257166,69981919,2694447797,126128146156,
%T A006846 7054258103921,464584757637001,35586641825705882,3136942184333040727,
%U A006846 315295985573234822561,35843594275585750890976,4575961401477587844760793,651880406652100451820206941
%N A006846 Hammersley's polynomial p_n(1).
%C A006846 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 A006846 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006846 Vincenzo Librandi, <a href="/A006846/b006846.txt">Table of n, a(n) for n = 0..100</a>
%H A006846 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 A006846 J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
%F A006846 a(n) = Sum_{k>=0} (-1)^(n+k)*A065547(n, k) = Sum_{k>=0} A085707(n, k). - _Philippe Deléham_, Feb 26 2004
%F A006846 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 A006846 a(n) = (-1)^n*A104027(n, 0). a(n+1) = (-1)^(n+1)*A104030(n, 0). - _Paul D. Hanna_, Mar 06 2005
%F A006846 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 A006846 a(n) ~ 4*cosh(sqrt(3)*Pi/2) * (2*n)! / Pi^(2*n+1). - _Vaclav Kotesovec_, Jun 07 2021
%p A006846 A006846 := proc(n)
%p A006846     option remember ;
%p A006846     if n =0 then
%p A006846         return 1;
%p A006846     else
%p A006846         add(binomial(2*n,2*m)*procname(m)/(-4)^(n-m),m=0..n-1) ;
%p A006846         (3/4)^n-% ;
%p A006846     end if
%p A006846 end proc:
%p A006846 seq(A006846(n),n=0..20) ; # _R. J. Mathar_, Jan 10 2018
%t A006846 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 A006846 (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 A006846 (Julia)
%o A006846 function A006846list(len::Int)  # Algorithm of L. Seidel (1877)
%o A006846     R = Array{BigInt}(len)
%o A006846     A = fill(BigInt(0), len+1); A[1] = 1
%o A006846     for n in 1:len
%o A006846         for k in n:-1:2 A[k] += A[k+1] end
%o A006846         for k in 2:1:n A[k] += A[k-1] end
%o A006846         R[n] = A[n]
%o A006846     end
%o A006846     return R
%o A006846 end
%o A006846 println(A006846list(20)) # _Peter Luschny_, Jan 02 2018
%Y A006846 Cf. A065547, A085707, A104027, A104030.
%K A006846 nonn
%O A006846 0,3
%A A006846 _N. J. A. Sloane_