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Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).
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%I #13 Feb 23 2023 14:11:58

%S 1,1,4,12,45,168,663,2667,10982,45925,194732,834912,3614063,15771795,

%T 69316740,306534564,1362986799,6089916936,27328613142,123118156260,

%U 556626199974,2524659817449,11484671681511,52384730922720,239534402969925,1097805759803893,5042014405418968

%N Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).

%C The g.f. for the Fibonacci numbers is 1/(1-x-x^2) and the g.f. M(x) for the Motzkin numbers satisfies: M(x) = 1 + x*M(x) + x^2*M(x)^2.

%e G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 45*x^4 + 168*x^5 + 663*x^6 +...

%e where A(x) = 1*1 + 1*1*x + 2*2*x^2 + 3*4*x^3 + 5*9*x^4 + 8*21*x^5 + 13*51*x^6 + 21*127*x^7 + 34*323*x^8 +...+ A000045(n+1)*A001006(n)*x^n +...

%o (PARI) {A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}

%o {a(n)=fibonacci(n+1)*A001006(n)}

%Y Cf. A098614, A200538, A200540, A098616.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 18 2011