A352237 - OEIS

%I #10 Nov 16 2023 12:54:20

%S 1,1,4,37,532,9994,226252,5910445,173581060,5634589906,199792389160,

%T 7671942375898,316936631324368,14011781050744984,660054967923455212,

%U 33008607551445324157,1746771084107236755604,97536010045722766992778,5731874036042145864368824

%N G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 3*x*A'(x)).

%H Vaclav Kotesovec, <a href="/A352237/b352237.txt">Table of n, a(n) for n = 0..378</a>

%F G.f. A(x) satisfies:

%F (1) [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1) for n >= 1.

%F (2) A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).

%F (3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (3*x).

%F (4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(3*x) dx ).

%F a(n) ~ c * 3^n * n! * n^(4/3), where c = 0.0543186200722307001992331... - _Vaclav Kotesovec_, Nov 16 2023

%e G.f.: A(x) = 1 + x + 4*x^2 + 37*x^3 + 532*x^4 + 9994*x^5 + 226252*x^6 + 5910445*x^7 + 173581060*x^8 + ...

%e such that A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).

%e Related table.

%e The table of coefficients of x^k in A(x)^(3*n+1) begins:

%e n=0: [1,  1,   4,   37,   532,   9994,   226252, ...];

%e n=1: [1,  4,  22,  200,  2717,  48788,  1069122, ...];

%e n=2: [1,  7,  49,  462,  6069, 104664,  2219784, ...];

%e n=3: [1, 10,  85,  850, 11020, 183832,  3777355, ...];

%e n=4: [1, 13, 130, 1391, 18083, 294203,  5869734, ...];

%e n=5: [1, 16, 184, 2112, 27852, 445632,  8659920, ...];

%e n=6: [1, 19, 247, 3040, 41002, 650161, 12353059, ...]; ...

%e in which the following pattern holds:

%e [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1), n >= 1,

%e as illustrated by

%e [x^1] A(x)^4 = 4 = [x^0] 4*A(x)^4 = 4*1;

%e [x^2] A(x)^7 = 49 = [x^1] 7*A(x)^7 = 7*7;

%e [x^3] A(x)^10 = 850 = [x^2] 10*A(x)^10 = 10*85;

%e [x^4] A(x)^13 = 18083 = [x^3] 13*A(x)^13 = 13*1391;

%e [x^5] A(x)^16 = 445632 = [x^4] 16*A(x)^16 = 16*27852;

%e [x^6] A(x)^19 = 12353059 = [x^5] 19*A(x)^19 = 19*650161; ...

%e Also, compare the above terms along the diagonal to the series

%e B(x) = A(x*B(x)^3) = 1 + x + 7*x^2 + 85*x^3 + 1391*x^4 + 27852*x^5 + 650161*x^6 + 17204220*x^7 + ...

%e where B(x)^3 = (1/x) * Series_Reversion( x/A(x)^3 ).

%o (PARI) /* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */

%o {a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 3*x*A' + x*O(x^n)) );

%o polcoeff(A,n)}

%o for(n=0,20, print1(a(n),", "))

%o (PARI) /* Using [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1)*A(x)^(3*n+1) */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = polcoeff((x*Ser(A)^(3*(#A)-2) - Ser(A)^(3*(#A)-2)/(3*(#A)-2)),#A-1));A[n+1]}

%o for(n=0,20, print1(a(n),", "))

%Y Cf. A088715, A286797, A317352, A352235, A352236, A352238.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 08 2022