A360234 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n >= 0.

1, 1, 4, 33, 414, 6750, 131963, 2957899, 73968136, 2027178710, 60143834893, 1914750144642, 64984397381766, 2339387034919340, 88976089246855623, 3563952072597604091, 149941204887915187568, 6610797722288579969347, 304837386103152855175255, 14675559490665539299350303

Offset

0,3

Links

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

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)^(n+2))^(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)^(n+2))^(n+1) / (n+1).

a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 3.146325060582260657459991059461810..., c = 0.007037477865521004701131626931596125... - Vaclav Kotesovec, Jan 31 2023

Example

G.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 414*x^4 + 6750*x^5 + 131963*x^6 + 2957899*x^7 + 73968136*x^8 + 2027178710*x^9 + 60143834893*x^10 + ...
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 + 46*x^3 + 603*x^4 + 10011*x^5 + 197357*x^6 + 4444483*x^7 + 111520277*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+2))^(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, 184/4, 3015/5, 60066/6, 1381499/7, 35555864/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  4,  33,  414,  6750,  131963,  2957899, ...];
n=1: [1, 2,  9,  74,  910, 14592,  281827,  6261048, ...];
n=2: [1, 3, 15, 124, 1500, 23673,  451690,  9944484, ...];
n=3: [1, 4, 22, 184, 2197, 34156,  643878, 14046740, ...];
n=4: [1, 5, 30, 255, 3015, 46221,  860965, 18610170, ...];
n=5: [1, 6, 39, 338, 3969, 60066, 1105794, 23681298, ...];
n=6: [1, 7, 49, 434, 5075, 75908, 1381499, 29311192, ...];
n=7: [1, 8, 60, 544, 6350, 93984, 1691528, 35555864, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+2))^(n+1):
n=0: [1, 1,   2,    9,    74,    910,   14592,   281827, ...];
n=1: [1, 2,   7,   36,   287,   3338,   51315,   963446, ...];
n=2: [1, 3,  15,   91,   744,   8337,  122662,  2227101, ...];
n=3: [1, 4,  26,  184,  1591,  17600,  249194,  4361112, ...];
n=4: [1, 5,  40,  325,  3015,  33656,  463710,  7824385, ...];
n=5: [1, 6,  57,  524,  5244,  60066,  816474, 13339956, ...];
n=6: [1, 7,  77,  791,  8547, 101619, 1381499, 22023891, ...];
n=7: [1, 8, 100, 1136, 13234, 164528, 2263888, 35555864, ...]; ...
to see that the main diagonals of the tables are the same.

Prog

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

Crossrefs

Cf. A360231, A302702, A302703, A360235, A360236, A360237.

Cf. A360346, A360338.

Sequence in context: A343673 A343686 A364980 * A111534 A162655 A216135

Adjacent sequences: A360231 A360232 A360233 * A360235 A360236 A360237

Keyword

nonn

Author

Paul D. Hanna, Jan 30 2023

Status

approved