A385912 A diagonal (unsigned) of triangle A385910; a(n) = -A385910(n+2,n) for n >= 1.

1, 8, 49, 296, 1815, 11284, 70924, 449616, 2869779, 18418400, 118749345, 768537120, 4990021764, 32489701776, 212048505160, 1386886206112, 9087724409547, 59646983740680, 392071446052195, 2580601721867400, 17005938279649935, 112190574812699460, 740878216459158960, 4897062582469861440, 32395964187696107700

Offset: 1

Author

Paul D. Hanna, Jul 14 2025

Keywords

• nonn

Comments

Triangle A385910 has g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).

The main diagonal of A385910 has g.f. G(x) = 1 + x*G(x)^3 (A001764).

Conjectures:

(C.1) a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 is the Fibbinary numbers.

(C.2) a(n) == 1 (mod 3) iff n = 3^k + 1 for k >= 0.

(C.3) a(n) == 2 (mod 3) iff n = A038464(k)/2 + 1 for k >= 1 or n = 1, where A038464 lists sums of 2 distinct powers of 3.

Examples

			G.f. A(x) = x + 8*x^2 + 49*x^3 + 296*x^4 + 1815*x^5 + 11284*x^6 + 70924*x^7 + 449616*x^8 + 2869779*x^9 + 18418400*x^10 + 118749345*x^11 + 768537120*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..180

Crossrefs

Cf. A385910, A003714, A118113, A038464.

Programs

PARI

\\ a(n) = -A385910(n+2,n) for n >= 1
{A385910(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 + 3*y*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 2*y*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(polcoef(-A385910(n+2),n),", "))