A200539 -id:A200539 - cp's OEIS frontend

A200538 Product of Jacobsthal and Motzkin numbers: a(n) = A001045(n+1)*A001006(n).

1, 1, 6, 20, 99, 441, 2193, 10795, 55233, 284735, 1494404, 7914270, 42360541, 228460935, 1241224182, 6784445340, 37288826697, 205937705799, 1142317727466, 6361104740100, 35548154733969, 199295884785459, 1120615326442269, 6318077793648075, 35710056983891367, 202297486497822121

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

The g.f. for the Jacobsthal numbers is 1/(1-x-2*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 + x + 6*x^2 + 20*x^3 + 99*x^4 + 441*x^5 + 2193*x^6 +...
where A(x) = 1*1 + 1*1*x + 3*2*x^2 + 5*4*x^3 + 11*9*x^4 + 21*21*x^5 + 43*51*x^6 + 85*127*x^7 + 171*323*x^8 +...+ A001045(n+1)*A001006(n)*x^n +...

Cf. A200375, A200539, A200540, A001045, A001006.

{A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}
{A001045(n)=polcoeff( x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=A001045(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

A200538 Product of Jacobsthal and Motzkin numbers: a(n) = A001045(n+1)*A001006(n).

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