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A302703 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+1))^(n+1) for n>=0. 2
1, 1, 3, 21, 235, 3470, 61933, 1274893, 29423331, 747440115, 20636072811, 613611700946, 19517927805840, 660667692682175, 23699856058131981, 897955765812058192, 35832679277251514074, 1502303284645831488072, 66031982339561373164915, 3036884343153028302140119, 145885192794643951791449387 (list; graph; refs; listen; history; text; internal format)
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+1))^(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+1))^(n+1) / (n+1).

a(n) ~ c * d^n * n! * n^alfa, where d = 2.12460658362428979..., alfa = 2.20132968515..., c = 0.026186121837... - Vaclav Kotesovec, Oct 06 2020

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 235*x^4 + 3470*x^5 + 61933*x^6 + 1274893*x^7 + 29423331*x^8 + 747440115*x^9 + 20636072811*x^10 + ...

RELATED SERIES.

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

B(x) = 1 + x + 4*x^2 + 31*x^3 + 356*x^4 + 5291*x^5 + 94592*x^6 + 1948763*x^7 + 45025516*x^8 + 1145651239*x^9 + 31696223593*x^10 + ... + b(n)*x^n + ...

such that b(n) = [x^n] (1 + x*A(x)^(n+1))^(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, 12/3, 124/4, 1780/5, 31746/6, 662144/7, 15590104/8, ...]

ILLUSTRATION OF DEFINITION.

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

n=0: [1, 1, 3, 21, 235, 3470, 61933, 1274893, ...];

n=1: [1, 2, 7, 48, 521, 7536, 132657, 2704342, ...];

n=2: [1, 3, 12, 82, 867, 12288, 213282, 4304877, ...];

n=3: [1, 4, 18, 124, 1283, 17828, 305056, 6094832,  ...];

n=4: [1, 5, 25, 175, 1780, 24271, 409380, 8094540, ...];

n=5: [1, 6, 33, 236, 2370, 31746, 527824, 10326546, ...];

n=6: [1, 7, 42, 308, 3066, 40397, 662144, 12815839, ...];

n=7: [1, 8, 52, 392, 3882, 50384, 814300, 15590104, ...]; ...

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

n=0: [1, 1, 1, 3, 21, 235, 3470, 61933, ...];

n=1: [1, 2, 5, 18, 114, 1166, 16355, 283142, ...];

n=2: [1, 3, 12, 55, 354, 3372, 44463, 739917, ...];

n=3: [1, 4, 22, 124, 857, 7908, 98244, 1558788, ...];

n=4: [1, 5, 35, 235, 1780, 16501, 195980, 2955095, ...];

n=5: [1, 6, 51, 398, 3321, 31746, 368032, 5294250, ...];

n=6: [1, 7, 70, 623, 5719, 57302, 662144, 9182013, ...];

n=7: [1, 8, 92, 920, 9254, 98088, 1149804, 15590104, ...]; ...

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+1))^(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. A302702.

Sequence in context: A078586 A179331 A138903 * A334262 A234855 A058562

Adjacent sequences:  A302700 A302701 A302702 * A302704 A302705 A302706

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 16 2018

STATUS

approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)