A383452 Column 3 in A104978.

0, 0, 0, 12, 165, 1430, 10010, 61880, 352716, 1899240, 9806280, 49031400, 239028075, 1141710570, 5362579950, 24837212400, 113678010600, 515030986800, 2312957340720, 10307744670600, 45626928615450, 200758485907980, 878623171119540, 3826892034209552, 16596215454480200, 71691488703052400, 308585103547921200, 1323929637802371600

Offset: 0

Author

N. J. A. Sloane, May 02 2025

Keywords

• nonn

Links

• N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See table on page 12.

Crossrefs

Cf. A104987.

Programs

Maple

a := n -> ifelse(n < 3, 0, (3 + 2*n)! / (6*(n - 3)!*(n + 4)!)): seq(a(n), n = 0..27);
y := sqrt(1 - 4*x): gf := (1/(2*(x*y)^4))*((210*x^4 - 420*x^3 + 252*x^2 - 60*x + 5)/y -(32*x^4 - 176*x^3 + 162*x^2 - 50*x + 5)): ser := series(gf, x, 34):
seq(coeff(ser, x, n), n = 0..27); # Peter Luschny, May 04 2025

Formula

From Peter Luschny, May 04 2025: (Start)

a(n) = (3 + 2*n)! / (6*(n - 3)!*(n + 4)!) for n >= 3.

a(n) = [x^n] (1/(2*(x*y)^4))*((210*x^4 - 420*x^3 + 252*x^2 - 60*x + 5)/y -(32*x^4 - 176*x^3 + 162*x^2 - 50*x + 5)) where y = sqrt(1 - 4*x). (End)