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A006846
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Hammersley's polynomial p_n(1).
(Formerly M1807)
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13
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1, 1, 2, 7, 41, 376, 5033, 92821, 2257166, 69981919, 2694447797, 126128146156, 7054258103921, 464584757637001, 35586641825705882, 3136942184333040727, 315295985573234822561, 35843594275585750890976, 4575961401477587844760793, 651880406652100451820206941
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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