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A300992 E.g.f. A(x) satisfies: [x^n] A(x)^(6*n) = (n+5) * [x^(n-1)] A(x)^(6*n) for n>=1. 7
1, 1, 3, 55, 2233, 141201, 12458731, 1435102663, 206465053425, 35963535971233, 7412714454497491, 1776535156724561751, 488255792062034106793, 152177253891382689328945, 53295007883395937033340603, 20811797234198326671764036071, 9002626614458116653486533691361, 4289501522632944577576478918096193, 2240137918573757743881572713997828515 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - 5*x*A'(x)) / (A(x) - 6*x*A'(x)) ).

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2233*x^4/4! + 141201*x^5/5! + 12458731*x^6/6! + 1435102663*x^7/7! + 206465053425*x^8/8! + 35963535971233*x^9/9! + ...

such that [x^n] A(x)^(6*n) = (n+5) * [x^(n-1)] A(x)^(6*n) for n>=1.

RELATED SERIES.

A(x)^6 = 1 + 6*x + 48*x^2/2! + 720*x^3/3! + 23328*x^4/4! + 1325376*x^5/5! + 109921536*x^6/6! + 12138398208*x^7/7! + 1692740643840*x^8/8! + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in A(x)^(6*n) begins:

n=1: [(1), (6), 24, 120, 972, 55224/5, 763344/5, ...];

n=2: [1, (12), (84), 528, 3960, 197568/5, 2494656/5, ...];

n=3: [1, 18, (180), (1440), 11556, 543672/5, 6306336/5, ...];

n=4: [1, 24, 312, (3072), (27648), 1313856/5, 14451264/5, ...];

n=5: [1, 30, 480, 5640, (57420), (574200), 6220080, ...];

n=6: [1, 36, 684, 9360, 107352, (5759424/5), (63353664/5), ...]; ...

in which the coefficients in parenthesis are related by

6 = 6*(1); 84 = 7*(12); 1440 = 8*(180); 27648 = 9*(3072); 574200 = 10*(57420); 63353664/5 = 11*(5759424/5); ...

illustrating: [x^n] A(x)^(6*n) = (n + 5) * [x^(n-1)] A(x)^(6*n).

LOGARITHMIC PROPERTY.

The logarithm of the e.g.f. is an integer power series in x satisfying

log(A(x)) = x * (1 - 5*x*A'(x)/A(x)) / (1 - 6*x*A'(x)/A(x));

explicitly,

log(A(x)) = x + x^2 + 8*x^3 + 84*x^4 + 1080*x^5 + 16056*x^6 + 266256*x^7 + 4816080*x^8 + 93638016*x^9 + 1937252160*x^10 + ... + A300993(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(6*(#A-1))); A[#A] = ((#A+4)*V[#A-1] - V[#A])/(6*(#A-1)) ); n!*polcoeff( Ser(A), n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) {a(n) = my(A=1); for(i=1, n, A = exp( x*(A-5*x*A')/(A-6*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A300993, A182962, A300735, A300986, A300988, A300990, A300992.

Sequence in context: A119192 A103134 A235534 * A304640 A015099 A297965

Adjacent sequences:  A300989 A300990 A300991 * A300993 A300994 A300995

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 19 2018

STATUS

approved

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Last modified November 14 12:44 EST 2019. Contains 329116 sequences. (Running on oeis4.)