A358092 Row sums of the convolution triangle of the Motzkin numbers (A202710).

1, 1, 3, 9, 28, 88, 279, 889, 2843, 9115, 29279, 94183, 303294, 977522, 3152709, 10173671, 32844544, 106073200, 342671109, 1107278239, 3578704532, 11568322736, 37400611581, 120931966547, 391065616195, 1264729338163, 4090528413309, 13230930776769, 42798305388298

Offset: 0

Author

Peter Luschny, Oct 29 2022

Keywords

• nonn

Crossrefs

Cf. A202710, A001006.

Programs

Maple

ogf := (sqrt(x + 1)*(1 - 2*x) + sqrt(1 - 3*x)) / (sqrt(x + 1)*(1 - 4*x) + sqrt(1 - 3*x)): ser := series(ogf, x, 32): seq(coeff(ser, x, n), n = 0..28);
# Alternatively:
a := proc(n) option remember; ifelse(n < 5, [1, 1, 3, 9, 28][n + 1],
((36-12*n)*a(n-4) + (30-14*n)*a(n-3) + (3*n-3)*a(n-2) + a(n-1)*(4*n-3))/n) end:
seq(a(n), n = 0..28);

Formula

a(n) = [x^n] (sqrt(x + 1)*(1 - 2*x) + sqrt(1 - 3*x)) / (sqrt(x + 1)*(1 - 4*x) + sqrt(1 - 3*x)).

a(n) = ((36-12*n)*a(n-4) + (30-14*n)*a(n-3) + (3*n-3)*a(n-2) + (4*n-3)*a(n-1))/n for n >= 5.