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A145349 G.f. satisfies: A(x/A(x)^3) = 1 + x*A(x).

%I A145349 #14 Jan 11 2016 11:08:56
%S A145349 1,1,4,34,416,6319,111124,2177346,46440184,1061938195,25762345804,
%T A145349 658072997702,17600573291712,490770914734054,14219015899154068,
%U A145349 426904437068035200,13252855203929697200,424634035832800883743,14020984603588221319072,476460483567807053745450
%N A145349 G.f. satisfies: A(x/A(x)^3) = 1 + x*A(x).
%C A145349 More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
%C A145349 A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
%C A145349 thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.
%F A145349 G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^3) and A(x) = G(x/A(x)^3).
%F A145349 a(n) = [x^(n-1)] 4*A(x)^(3*n+1)/(3*n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 4*A(x)^(3*n+1)/(3*n+1) for n>=1 (see comment).
%e A145349 G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 416*x^4 + 6319*x^5 +...
%e A145349 A(x)^3 = 1 + 3*x + 15*x^2 + 127*x^3 + 1512*x^4 + 22419*x^5 +...
%e A145349 A(x/A(x)^3) = 1 + x + x^2 + 4*x^3 + 34*x^4 + 416*x^5 + 6319*x^6 +...
%e A145349 A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^3):
%e A145349 G(x) = 1 + x + 7*x^2 + 82*x^3 + 1239*x^4 + 21942*x^5 + 434746*x^6 +...
%e A145349 G(x)^3 = 1 + 3*x + 24*x^2 + 289*x^3 + 4377*x^4 + 77097*x^5 +...
%e A145349 To illustrate the formula a(n) = [x^(n-1)] 4*A(x)^(3*n+1)/(3*n+1),
%e A145349 form a table of coefficients in A(x)^(3*n+1) as follows:
%e A145349 A^4: [(1), 4, 22, 188, 2217, 32516, 555972, ...];
%e A145349 A^7: [1, (7), 49, 441, 5131, 73248, 1220457, ...];
%e A145349 A^10: [1, 10, (85), 820, 9590, 134482, 2191060, ...];
%e A145349 A^13: [1, 13, 130, (1352), 16107, 223886, 3582072, ...];
%e A145349 A^16: [1, 16, 184, 2064, (25276), 351072, 5541912, ...];
%e A145349 A^19: [1, 19, 247, 2983, 37772, (527839), 8260174, ...];  ...
%e A145349 in which the main diagonal forms the initial terms of this sequence:
%e A145349 [4/4*(1), 4/7*(7), 4/10*(85), 4/13*(1352), 4/16*(25276), 4/19*(527839), ...].
%e A145349 ALTERNATE GENERATING METHOD.
%e A145349 This sequence forms column zero in the follow array.
%e A145349 Let A denote this sequence, and A^3 the self-convolution cube of A.
%e A145349 Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A^3 and the remaining terms in a given row to obtain the next row:
%e A145349 [1, 1, 1, 4, 34, 416, 6319, 111124, 2177346, 46440184, 1061938195, ...];
%e A145349 [1, 4, 22, 188, 2217, 32516, 555972, 10655628, 223313220, 5034249556, ...];
%e A145349 [4, 34, 314, 3619, 50829, 833591, 15417781, 313704516, 6900409869, ...];
%e A145349 [34, 416, 5071, 70714, 1131649, 20377616, 404581945, 8712077584, ...];
%e A145349 [416, 6319, 92167, 1472688, 26106282, 508663862, 10756749655, ...];
%e A145349 [6319, 111124, 1843974, 32709364, 630332565, 13142389012, 294045605744, ...];
%e A145349 [111124, 2177346, 39908146, 770233015, 15926231353, 352361228127, ...];
%e A145349 [46440184, 1061938195, 22576531219, 497127293708, 11532665230502, ...]; ...
%o A145349 (PARI) {a(n)=local(A=1+x,G);for(i=0,n,G=(serreverse(x/(A+x*O(x^n))^3)/x)^(1/3);A=1+x*G^4);polcoeff(A,n)}
%o A145349 (PARI) /* This sequence is generated when k=3, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */
%o A145349 {a(n, k=3, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
%o A145349 for(n=0,20,print1(a(n),", "))
%o A145349 (PARI) /* Prints terms 0..30 */
%o A145349 {A=[1];
%o A145349 for(m=1,30,
%o A145349   B=A;
%o A145349   for(i=1,m-1, C=Vec(Ser(A)^3*Ser(B)); B=vector(#C-1,n,C[n+1]) );
%o A145349   A=concat(A,0);A[#A]=B[1]
%o A145349 );
%o A145349 A} \\ _Paul D. Hanna_, Jan 10 2016
%Y A145349 Cf. variants: A002293, A145347, A145348, A120974,  A145345, A145350, A147664.
%K A145349 nonn
%O A145349 0,3
%A A145349 _Paul D. Hanna_, Nov 11 2008