A291892 Number of symmetrically unique Dyck paths of semilength n and height eight.

1, 8, 71, 461, 2706, 14235, 70161, 327469, 1469596, 6389144, 27097948, 112630404, 460511702, 1857372265, 7406424903, 29250500171, 114576069911, 445647539540, 1722814022086, 6624828067482, 25356181172529, 96650473757117, 367059233827762, 1389476976680608

Offset

8,2

Links

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Index entries for linear recurrences with constant coefficients, signature (15, -84, 178, 160, -1436, 1985, 1447, -6220, 4096, 3445, -5665, 1925, 405, -275, 25).

Formula

G.f.: x^8*(-1+7*x-35*x^2+110*x^3-171*x^4+113*x^5-25*x^6+5*x^7) / ((x-1) *(x^2-3*x+1) *(5*x^2-5*x+1) *(x^3+3*x^2-1) *(x^3-9*x^2+6*x-1) *(5*x^4-5*x^2+1)).

a(n) = 15*a(n-1) - 84*a(n-2) + 178*a(n-3) + 160*a(n-4) - 1436*a(n-5) + 1985*a(n-6) + 1447*a(n-7) - 6220*a(n-8) + 4096*a(n-9) + 3445*a(n-10) - 5665*a(n-11) + 1925*a(n-12) + 405*a(n-13) - 275*a(n-14) + 25*a(n-15). - Wesley Ivan Hurt, Feb 08 2026

Mathematica

CoefficientList[Series[(-1 + 7*x - 35*x^2 + 110*x^3 - 171*x^4 + 113*x^5 - 25*x^6 + 5*x^7)/((x - 1)*(x^2 - 3*x + 1)*(5*x^2 - 5*x + 1)*(x^3 + 3*x^2 - 1)*(x^3 - 9*x^2 + 6*x - 1)*(5*x^4 - 5*x^2 + 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Feb 08 2026 *)

Crossrefs

Column k=8 of A291883.

Sequence in context: A043086 A003364 A376053 * A217328 A193102 A390203

Adjacent sequences: A291889 A291890 A291891 * A291893 A291894 A291895

Keyword

nonn,easy

Author

Alois P. Heinz, Sep 05 2017

Status

approved