A360344 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n))^(n+1) for n >= 0.

1, 1, 4, 45, 820, 19820, 582007, 19812744, 760177656, 32275309743, 1497313010037, 75208566398988, 4062020902196139, 234638046113989856, 14432573619909530980, 941883830760366274935, 65013065172020161949992, 4733236746727327140204578, 362575149419405494321544263

Offset: 0

Author

Paul D. Hanna, Feb 05 2023

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 45*x^3 + 820*x^4 + 19820*x^5 + 582007*x^6 + 19812744*x^7 + 760177656*x^8 + 32275309743*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 5*x^2 + 58*x^3 + 1057*x^4 + 25471*x^5 + 746143*x^6 + 25364298*x^7 + 972602305*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 15/3, 232/4, 5285/5, 152826/6, 5223001/7, 202914384/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  4,  45,   820,  19820,  582007,  19812744, ...];
n=1: [1, 2,  9,  98,  1746,  41640, 1212239,  41021862, ...];
n=2: [1, 3, 15, 160,  2790,  65643, 1894300,  63714729, ...];
n=3: [1, 4, 22, 232,  3965,  92028, 2632070,  87984416, ...];
n=4: [1, 5, 30, 315,  5285, 121011, 3429725, 113930075, ...];
n=5: [1, 6, 39, 410,  6765, 152826, 4291758, 141657348, ...];
n=6: [1, 7, 49, 518,  8421, 187726, 5223001, 171278801, ...];
n=7: [1, 8, 60, 640, 10270, 225984, 6228648, 202914384, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n))^(n+1):
n=0: [1, 1,   0,    0,     0,      0,       0,         0, ...];
n=1: [1, 2,   5,   22,   218,   3724,   87245,   2516506, ...];
n=2: [1, 3,  15,   91,   888,  13929,  308182,   8594133, ...];
n=3: [1, 4,  30,  232,  2397,  35712,  742902,  19860536, ...];
n=4: [1, 5,  50,  470,  5285,  77631, 1530000,  38965400, ...];
n=5: [1, 6,  75,  830, 10245, 152826, 2900808,  70300080, ...];
n=6: [1, 7, 105, 1337, 18123, 280140, 5223001, 121085308, ...];
n=7: [1, 8, 140, 2016, 29918, 485240, 9053576, 202914384, ...]; ...
to see that the main diagonals of the tables are the same.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A360342, A360343, A360345, A360346, A360347.

Cf. A302702, A302703.

Programs

PARI

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

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

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

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

a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 1.1169011279372..., c = 0.052820142023857... - Vaclav Kotesovec, Feb 06 2023