A062747 Row sums of (unsigned) staircase array A062746.
1, 7, 89, 1447, 26713, 532391, 11165785, 242851751, 5427716185, 123901026215, 2876525797465, 67710590623655, 1612262780199001, 38764533106581415, 939825790848884825, 22950405085586497447
Offset: 0
Formula
a(n)=N(3; k, x=-1), with the polynomials N(3; k, x) from the staircase array A062746.
a(n) = 2*( Sum_{j = 0..n} (-1)^j*C(3; n-j)*4^(n-j) ) - (-1)^n with C(3; n) := A001764(n) = A062993(n+1, 1) (a Pfaff-Fuss or 3-Raney sequence).
G.f.: (2*c(3; 4*x)-1)/(1+x) with c(3; x)= RootOf(x*A001764%20%5Bformula%20for%20a(n)%20and%20g.f.%20corrected%20by%20_Peter%20Bala">Z^3-_Z +1), the g.f. of A001764 [formula for a(n) and g.f. corrected by _Peter Bala, Mar 26 2020].
Conjectural recurrence: n*(2*n+1)*a(n) = (4*n-3)*(13*n-4)*a(n-1) + 6*(3*n-1)*(3*n-2)*a(n-2) with a(0) = 1, a(1) = 7. - Peter Bala, Mar 25 2020