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A302702 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^n)^(n+1) for n>=0. 3
1, 1, 2, 12, 120, 1595, 25823, 485254, 10278756, 240814116, 6159248281, 170371486813, 5060981349876, 160573684489465, 5417789356278015, 193693975380448414, 7315287863625954712, 291082028021247460862, 12174286414586563087259, 534059044249856004891501, 24524697505864740171996008 (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)^(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 = 2.12460658362428979..., alfa = 1.25633430971..., c = 0.08016154855... - Vaclav Kotesovec, Oct 06 2020

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. A302703, A303062.

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

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Last modified July 23 14:44 EDT 2021. Contains 346259 sequences. (Running on oeis4.)