A360343 - OEIS

A360343

G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) for n >= 0.

%I #9 Feb 06 2023 04:10:27

%S 1,1,3,31,526,11907,328980,10580531,384937042,15549217485,

%T 688430225102,33096289502982,1715499922758709,95339852384471586,

%U 5655337634718941111,356683962066445400017,23840465113068534382248,1683771696557415075462436,125327912444852044066759399

%N G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) for n >= 0.

%H Paul D. Hanna, <a href="/A360343/b360343.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) for n>=0.

%F (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) / (n+1).

%F a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 0.5984002265754..., c = 0.08321697608093... - _Vaclav Kotesovec_, Feb 06 2023

%e G.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 526*x^4 + 11907*x^5 + 328980*x^6 + 10580531*x^7 + 384937042*x^8 + 15549217485*x^9 + ...

%e RELATED SERIES.

%e G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:

%e B(x) = 1 + x + 4*x^2 + 41*x^3 + 687*x^4 + 15433*x^5 + 424524*x^6 + 13620842*x^7 + 495005025*x^8 + ... + b(n)*x^n + ...

%e such that b(n) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) / (n+1),

%e as well as b(n) = [x^n] A(x)^(n+1) / (n+1),

%e so that {b(n)} begins:

%e [1/1, 2/2, 12/3, 164/4, 3435/5, 92598/6, 2971668/7, 108966736/8, ...].

%e ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, 1, 3, 31, 526, 11907, 328980, 10580531, ...];

%e n=1: [1, 2, 7, 68, 1123, 25052, 685891, 21923076, ...];

%e n=2: [1, 3, 12, 112, 1800, 39555, 1072896, 34076544, ...];

%e n=3: [1, 4, 18, 164, 2567, 55548, 1492336, 47093172, ...];

%e n=4: [1, 5, 25, 225, 3435, 73176, 1946745, 61028770, ...];

%e n=5: [1, 6, 33, 296, 4416, 92598, 2438866, 75942984, ...];

%e n=6: [1, 7, 42, 378, 5523, 113988, 2971668, 91899578, ...];

%e n=7: [1, 8, 52, 472, 6770, 137536, 3548364, 108966736, ...]; ...

%e Compare to the table of coefficients in (1 + x*A(x)^(2*n-1))^(n+1):

%e n=0: [1, 1, -1, -2, -26, -463, -10778, -303048, ...];

%e n=1: [1, 2, 3, 8, 69, 1120, 24937, 683012, ...];

%e n=2: [1, 3, 12, 55, 444, 6351, 132492, 3504654, ...];

%e n=3: [1, 4, 26, 164, 1411, 18560, 357624, 9024812, ...];

%e n=4: [1, 5, 45, 360, 3435, 43926, 785715, 18700710, ...];

%e n=5: [1, 6, 69, 668, 7134, 92598, 1570420, 35086104, ...];

%e n=6: [1, 7, 98, 1113, 13279, 179816, 2971668, 62645353, ...];

%e n=7: [1, 8, 132, 1720, 22794, 327032, 5403036, 108966736, ...]; ...

%e to see that the main diagonals of the tables are the same.

%o (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m-1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}

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

%Y Cf. A360342, A360344, A360345, A360346, A360347.

%Y Cf. A360231, A302702.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 05 2023