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A349048
G.f. A(x) satisfies: A(x) = 1 / (1 - x + x^4 * A(x)).
3
1, 1, 1, 1, 0, -2, -5, -9, -12, -10, 3, 35, 91, 163, 215, 163, -136, -858, -2107, -3675, -4639, -2879, 5161, 23741, 54910, 91988, 108843, 47483, -186582, -700420, -1527461, -2440985, -2656442, -507076, 6617735, 21456279, 44213835, 67037683, 65541879, -9699085, -232548686
OFFSET
0,6
LINKS
FORMULA
G.f.: (-1 + x + sqrt((1 - x)^2 + 4*x^4)) / (2*x^4).
a(0) = 1; a(n) = a(n-1) - Sum_{k=0..n-4} a(k) * a(n-k-4).
a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-2*k,2*k) * Catalan(k).
a(n) = F([(1-n)/4, (2-n)/4, (3-n)/4, -n/4], [2, (1-n)/2, -n/2], -64), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 07 2021
MATHEMATICA
nmax = 40; A[_] = 0; Do[A[x_] = 1/(1 - x + x^4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] - Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 40}]
Table[Sum[(-1)^k Binomial[n - 2 k, 2 k] CatalanNumber[k], {k, 0, Floor[n/4]}], {n, 0, 40}]
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Nov 06 2021
STATUS
approved