|
%I #13 Nov 28 2017 00:45:03
%S 1,1,3,17,110,783,5908,46433,376029,3115941,26293410,225166050,
%T 1951877304,17094430060,151026790086,1344405191931,12046557221374,
%U 108569555036541,983512740523989,8950335155129326,81786796816686222,750133309106091800,6903286130025559800,63724450582843480092,589897018021520290940,5474784437150040712036,50932035767512193052753,474865449870890392910894
%N G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).
%C Compare to: G(x)^3 = 1+x + x*(G(x)^2 + G(x)^4) holds when G(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).
%H Paul D. Hanna, <a href="/A295808/b295808.txt">Table of n, a(n) for n = 0..520</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 1 + Series_Reversion( x/(1 + 3*x + 8*x^2 + 11*x^3 + 10*x^4 + 5*x^5 + x^6) ).
%F (2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^3 - x^5 + x^6).
%F (3) A(x) satisfies: A(x) = 1/A(-x*A(x)^5).
%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 110*x^4 + 783*x^5 + 5908*x^6 + 46433*x^7 + 376029*x^8 + 3115941*x^9 + 26293410*x^10 + 225166050*x^11 + 1951877304*x^12 +...
%e such that A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).
%e RELATED SERIES.
%e A(x)^3 = 1 + 3*x + 12*x^2 + 70*x^3 + 468*x^4 + 3393*x^5 + 25932*x^6 + 205716*x^7 + 1677804*x^8 + 13980710*x^9 + 118505772*x^10 +...
%e A(x)^4 = 1 + 4*x + 18*x^2 + 108*x^3 + 735*x^4 + 5388*x^5 + 41496*x^6 + 331036*x^7 + 2711511*x^8 + 22670964*x^9 + 192695140*x^10 +...
%e A(x)^8 = 1 + 8*x + 52*x^2 + 360*x^3 + 2658*x^4 + 20544*x^5 + 164220*x^6 + 1346768*x^7 + 11269199*x^8 + 95834808*x^9 + 825905828*x^10 +...
%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^4 + x*A^8 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 3*x + 8*x^2 + 11*x^3 + 10*x^4 + 5*x^5 + x^6 +x*O(x^n)) ); polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A295809, A295810. A259757, A295537, A295538.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 27 2017
| 