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

1,6

Comments

Row sums of triangle A202327. - Peter Luschny, Apr 26 2017

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

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 *)

Prog

(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
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 41);
Coefficients(R!( (-1+x+Sqrt(1+2*x+5*x^2))/(2*(1+x)) )); // G. C. Greubel, Oct 20 2023
(SageMath)
def A108624_list(prec):
    P.<x> = 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

Crossrefs

Cf. A039980, A202327.

Except for signs, same as A108623.

Sequence in context: A137503 A320263 A215330 * A108623 A248248 A159550

Adjacent sequences: A108621 A108622 A108623 * A108625 A108626 A108627

Keyword

sign

Author

Christian G. Bower, Jun 12 2005

Status

approved