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

%I A385912 #9 Jul 20 2025 15:41:39
%S A385912 1,8,49,296,1815,11284,70924,449616,2869779,18418400,118749345,
%T A385912 768537120,4990021764,32489701776,212048505160,1386886206112,
%U A385912 9087724409547,59646983740680,392071446052195,2580601721867400,17005938279649935,112190574812699460,740878216459158960,4897062582469861440,32395964187696107700
%N A385912 A diagonal (unsigned) of triangle A385910; a(n) = -A385910(n+2,n) for n >= 1.
%C A385912 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).
%C A385912 The main diagonal of A385910 has g.f. G(x) = 1 + x*G(x)^3 (A001764).
%C A385912 Conjectures:
%C A385912 (C.1) a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 is the Fibbinary numbers.
%C A385912 (C.2) a(n) == 1 (mod 3) iff n = 3^k + 1 for k >= 0.
%C A385912 (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.
%H A385912 Paul D. Hanna, <a href="/A385912/b385912.txt">Table of n, a(n) for n = 1..180</a>
%e A385912 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 + ...
%o A385912 (PARI) \\ a(n) = -A385910(n+2,n) for n >= 1
%o A385912 {A385910(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
%o A385912 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]}
%o A385912 for(n=1, 25, print1(polcoef(-A385910(n+2),n),", "))
%Y A385912 Cf. A385910, A003714, A118113, A038464.
%K A385912 nonn
%O A385912 1,2
%A A385912 _Paul D. Hanna_, Jul 14 2025