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#29 by Paul D. Hanna at Mon Mar 11 10:01:36 EDT 2024
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(4) R(x^3) = R(x)^3/(1 - 3*R(x)) where R(A(x)) = x. - Paul D. Hanna, Mar 11 2024
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editing
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#28 by Paul D. Hanna at Mon Mar 11 09:53:12 EDT 2024
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| FORMULA
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(4) R(x^3) = R(x)^3/(1 - 3*R(x)) where R(A(x)) = x. - Paul D. Hanna, Mar 11 2024
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approved
editing
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#27 by Michael De Vlieger at Mon Mar 11 08:29:54 EDT 2024
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#26 by Joerg Arndt at Mon Mar 11 04:12:46 EDT 2024
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#25 by Paul D. Hanna at Mon Mar 11 03:07:21 EDT 2024
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#24 by Paul D. Hanna at Mon Mar 11 03:06:32 EDT 2024
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| FORMULA
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(3) A( x/(1+x+x^2) )^3 = A( x^3/(1+x+-x^23)^2 ). - Paul D. Hanna, Mar 11 2024
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#23 by Paul D. Hanna at Mon Mar 11 02:51:54 EDT 2024
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#22 by Paul D. Hanna at Sun Mar 10 17:29:31 EDT 2024
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#21 by Paul D. Hanna at Sun Mar 10 17:29:27 EDT 2024
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| EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 + 58295*x^13 + 174175*x^14 +... + ...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 270*x^8 + 839*x^9 + 2610*x^10 + 8127*x^11 + 25331*x^12 + 79035*x^13 + 246852*x^14 + 771808*x^15 +... + ...
A( x/(1+x+x^2) ) = x + x^4 + 2*x^7 + 6*x^10 + 22*x^13 + 88*x^16 + 367*x^19 + 1570*x^22 + 6843*x^25 + 30271*x^28 + 135530*x^31 + 612852*x^34 + 2794412*x^37 + 12832472*x^40 +... + ...
B(x) = 1 + x + x^2 + x^3 - x^5 - x^6 + 2*x^8 + 3*x^9 - 6*x^11 - 9*x^12 + 20*x^14 + 30*x^15 - 71*x^17 - 110*x^18 + 267*x^20 + 419*x^21 - 1041*x^23 +... + ...
C0(x) = 1 + x^3 - x^6 + 3*x^9 - 9*x^12 + 30*x^15 - 110*x^18 + 419*x^21 - 1648*x^24 + 6652*x^27 - 27369*x^30 + 114384*x^33 - 484276*x^36 +... + ...
C2(x) = ) = x^2 - x^5 + 2*x^8 - 6*x^11 + 20*x^14 - 71*x^17 + 267*x^20 - 1041*x^23 + 4168*x^26 - 17047*x^29 + 70902*x^32 - 298967* + ... + (-1)^(n-1)*A370446(n)*x^35 + 1275141^(3*x^38 +...n-1) + ...
then C2C0(x) = x^2/C0C2(x).
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| CROSSREFS
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Cf. A264229, A264230, A361763, A370446.
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#20 by Paul D. Hanna at Sun Mar 10 17:25:52 EDT 2024
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then C2(x) = x^2/C0(x);).
further, C2(A(x)) / A(x) = A(x) / C0(A(x)) = M(x), where M(x) is a g.f. of Motzkin numbers: M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).
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approved
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