A108624 G.f. satisfies x = A(x)*(1+A(x))/(1-A(x)-(A(x))^2).
1, 0, -1, 1, 1, -4, 3, 8, -23, 10, 67, -153, 9, 586, -1081, -439, 5249, -7734, -7941, 47501, -53791, -105314, 430119, -343044, -1249799, 3866556, -1730017, -13996097, 34243897, -1947204, -150962373, 296101864, 121857185
Offset: 1
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
Programs
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Julia
function A108624_list(len::Int) len <= 0 && return BigInt[] T = zeros(BigInt, len, len); T[1,1] = 1 S = Array(BigInt, len); S[1] = 1 for n in 2:len T[n,n] = 1 for k in 1:n-1 T[n,k] = (k > 1 ? T[n-1,k-1] : 0) - T[n-1,k] - T[n-1,k+1] end S[n] = sum(T[n,k] for k in 1:n) end S end println(A108624_list(33)) # Peter Luschny, Apr 27 2017
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Magma
R
:=PowerSeriesRing(Rationals(), 41); Coefficients(R!( (-1+x+Sqrt(1+2*x+5*x^2))/(2*(1+x)) )); // G. C. Greubel, Oct 20 2023 -
Mathematica
a[n_]:= Sum[k Sum[(-1)^(j-k) Binomial[j, 2j-n-k] Binomial[n, j], {j, 0, n}], {k, 1, n}]/n; Array[a, 33] (* Jean-François Alcover, Jun 13 2019, after Vladimir Kruchinin *) Rest@CoefficientList[Series[(-1+x+Sqrt[1+2*x+5*x^2])/(2*(1+x)), {x,0, 41}], x] (* G. C. Greubel, Oct 20 2023 *) -
SageMath
def A108624_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( (-1+x+sqrt(1+2*x+5*x^2))/(2*(1+x))).list() a=A108624_list(41); a[1:] # G. C. Greubel, Oct 20 2023
Formula
a(n) = (1/n)*Sum_{k=1..n} ( k * Sum_{j=0..n} (-1)^(k+j)*binomial(j, 2*j-n-k)*binomial(n,j) ). - Vladimir Kruchinin, May 19 2012
G.f.: (-1 + x + sqrt(1+2*x+5*x^2))/(2*(1+x)). - G. C. Greubel, Oct 20 2023
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