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Row sums of the convolution triangle of the Motzkin numbers (A202710).
1

%I #14 Oct 30 2022 03:06:26

%S 1,1,3,9,28,88,279,889,2843,9115,29279,94183,303294,977522,3152709,

%T 10173671,32844544,106073200,342671109,1107278239,3578704532,

%U 11568322736,37400611581,120931966547,391065616195,1264729338163,4090528413309,13230930776769,42798305388298

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

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

%F 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.

%p 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);

%p # Alternatively:

%p a := proc(n) option remember; ifelse(n < 5, [1, 1, 3, 9, 28][n + 1],

%p ((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:

%p seq(a(n), n = 0..28);

%Y Cf. A202710, A001006.

%K nonn

%O 0,3

%A _Peter Luschny_, Oct 29 2022