A302702 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^n)^(n+1) for n>=0.

1, 1, 2, 12, 120, 1595, 25823, 485254, 10278756, 240814116, 6159248281, 170371486813, 5060981349876, 160573684489465, 5417789356278015, 193693975380448414, 7315287863625954712, 291082028021247460862, 12174286414586563087259, 534059044249856004891501, 24524697505864740171996008

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

a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 1.256334309718765863868089027485828533429844901971596190707510781..., c = 0.080161548550419985236395573058502044572123359124998971614... - Vaclav Kotesovec, Oct 06 2020, updated Feb 05 2023

Example

G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 120*x^4 + 1595*x^5 + 25823*x^6 + 485254*x^7 + 10278756*x^8 + 240814116*x^9 + 6159248281*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 3*x^2 + 19*x^3 + 189*x^4 + 2496*x^5 + 40216*x^6 + 753775*x^7 + 15956057*x^8 + 374080591*x^9 + 6159248281*x^10 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^n)^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1, 2/2, 9/3, 76/4, 945/5, 14976/6, 281512/7, 6030200/8, ...]
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  2,  12,  120,  1595,  25823,  485254, ...];
n=1: [1, 2,  5,  28,  268,  3478,  55460, 1031414, ...];
n=2: [1, 3,  9,  49,  450,  5697,  89423, 1645281, ...];
n=3: [1, 4, 14,  76,  673,  8308, 128296, 2334456, ...];
n=4: [1, 5, 20, 110,  945, 11376, 172745, 3107440, ...];
n=5: [1, 6, 27, 152, 1275, 14976, 223529, 3973746, ...];
n=6: [1, 7, 35, 203, 1673, 19194, 281512, 4944024, ...];
n=7: [1, 8, 44, 264, 2150, 24128, 347676, 6030200, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^n)^(n+1):
n=0: [1, 1,  0,   0,    0,     0,      0,       0, ...];
n=1: [1, 2,  3,   6,   29,   268,   3458,   55124, ...];
n=2: [1, 3,  9,  28,  132,  1059,  12605,  192579, ...];
n=3: [1, 4, 18,  76,  395,  2940,  31872,  459048, ...];
n=4: [1, 5, 30, 160,  945,  6986,  70100,  940180, ...];
n=5: [1, 6, 45, 290, 1950, 14976, 143807, 1796430, ...];
n=6: [1, 7, 63, 476, 3619, 29589, 281512, 3321571, ...];
n=7: [1, 8, 84, 728, 6202, 54600, 529116, 6030200, ...]; ...
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)^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A360231, A303062, A302703, A360234, A360235, A360236, A360237.

Sequence in context: A052580 A134095 A204042 * A189981 A326000 A245067

Adjacent sequences: A302699 A302700 A302701 * A302703 A302704 A302705

Keyword

nonn

Author

Paul D. Hanna, Apr 16 2018

Status

approved