A349047 G.f. A(x) satisfies: A(x) = 1 / (1 - x + x^3 * A(x)).

1, 1, 1, 0, -2, -5, -7, -4, 10, 38, 70, 68, -40, -329, -767, -1012, -214, 2842, 8642, 14332, 10136, -21622, -96578, -196412, -213080, 96264, 1037344, 2608788, 3698996, 1121127, -10234567, -33425980, -58537486, -45735382, 83471346, 408899204, 871127040

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: (-1 + x + sqrt((1 - x)^2 + 4*x^3)) / (2*x^3).

a(0) = 1; a(n) = a(n-1) - Sum_{k=0..n-3} a(k) * a(n-k-3).

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-k,2*k) * Catalan(k).

a(n) = F([(1-n)/3, (2-n)/3, -n/3], [2, -n], -27), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 07 2021

Mathematica

nmax = 36; A[_] = 0; Do[A[x_] = 1/(1 - x + x^3 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 - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 36}]
Table[Sum[(-1)^k Binomial[n - k, 2 k] CatalanNumber[k], {k, 0, Floor[n/3]}], {n, 0, 36}]

Crossrefs

Cf. A000108, A007440, A023431, A100223, A214649, A343773, A349048.

Sequence in context: A146098 A296567 A205113 * A098486 A138320 A199275

Adjacent sequences: A349044 A349045 A349046 * A349048 A349049 A349050

Keyword

sign

Author

Ilya Gutkovskiy, Nov 06 2021

Status

approved