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 A006846 Hammersley's polynomial p_n(1). (Formerly M1807) 13
 1, 1, 2, 7, 41, 376, 5033, 92821, 2257166, 69981919, 2694447797, 126128146156, 7054258103921, 464584757637001, 35586641825705882, 3136942184333040727, 315295985573234822561, 35843594275585750890976, 4575961401477587844760793, 651880406652100451820206941 (list; graph; refs; listen; history; text; internal format)
 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. A065547, A085707, 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

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Last modified September 17 16:03 EDT 2024. Contains 375987 sequences. (Running on oeis4.)