A200538 -id:A200538 - cp's OEIS frontend

A200539 Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).

1, 1, 4, 12, 45, 168, 663, 2667, 10982, 45925, 194732, 834912, 3614063, 15771795, 69316740, 306534564, 1362986799, 6089916936, 27328613142, 123118156260, 556626199974, 2524659817449, 11484671681511, 52384730922720, 239534402969925, 1097805759803893, 5042014405418968

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

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.

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

Cf. A098614, A200538, A200540, A098616.

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

A200540 Product of Pell and Motzkin numbers: a(n) = A000129(n+1)*A001006(n).

Original entry on oeis.org

1, 2, 10, 48, 261, 1470, 8619, 51816, 318155, 1985630, 12561308, 80360280, 519013571, 3379514970, 22161470850, 146227235904, 970126550763, 6467496499590, 43304243215638, 291087137589552, 1963598081845335, 13288619577124122, 90195242361688863, 613843707553183800

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

The g.f. for the Pell numbers is 1/(1-2*x-x^2) and the g.f. M(x) for the Motzkin numbers satisfy: M(x) = 1 + x*M(x) + x^2*M(x)^2.

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 48*x^3 + 261*x^4 + 1470*x^5 + 8619*x^6 +...
where A(x) = 1*1 + 2*1*x + 5*2*x^2 + 12*4*x^3 + 29*9*x^4 + 70*21*x^5 + 169*51*x^6 + 408*127*x^7 + 985*323*x^8 +...+ A000129(n+1)*A001006(n)*x^n +...

Cf. A098616, A200538, A200539, A098614.

{A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}
{A000129(n)=polcoeff( x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=A000129(n+1)*A001006(n)}

cp's OEIS Frontend

A200539 Product of Fibonacci and Motzkin numbers: a(n) = A000045(n+1)*A001006(n).

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