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A300988 E.g.f. A(x) satisfies: [x^n] A(x)^(4*n) = (n + 3) * [x^(n-1)] A(x)^(4*n) for n>=1. 7
1, 1, 3, 43, 1369, 69561, 4991371, 471516403, 56029153713, 8112993527089, 1398528216254611, 281935928284459131, 65543089930613822473, 17373185629100099938153, 5201713100466658289659419, 1745470558150260528082445251, 652016607740826946854349450081, 269558306371535265856134699842913, 122707064351998882900943162086492963, 61225312946191234549695844364141862859 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare to: [x^n] exp(x)^(4*n) = 4 * [x^(n-1)] exp(x)^(4*n) for n>=1.

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) - 3*x*A'(x)) / (A(x) - 4*x*A'(x)) ).

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1369*x^4/4! + 69561*x^5/5! + 4991371*x^6/6! + 471516403*x^7/7! + 56029153713*x^8/8! + 8112993527089*x^9/9! + ...

such that [x^n] A(x)^(4*n) = (n+3) * [x^(n-1)] A(x)^(4*n) for n>=1.

RELATED SERIES.

A(x)^4 = 1 + 4*x + 24*x^2/2! + 304*x^3/3! + 8320*x^4/4! + 390144*x^5/5! + 26653696*x^6/6! + 2434011136*x^7/7! + 282056564736*x^8/8! + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in A(x)^(4*n) begins:

n=1: [(1), (4), 12, 152/3, 1040/3, 16256/5, 1665856/45, 152125696/315, ...];

n=2: [1, (8), (40), 592/3, 3728/3, 157376/15, 4992064/45, 86636800/63, ...];

n=3: [1, 12, (84), (504), 3264, 129408/5, 1273536/5, 104486784/35, ...];

n=4: [1, 16, 144, (3104/3), (21728/3), 283264/5, 23764096/45, 1844359168/315, ...];

n=5: [1, 20, 220, 5560/3, (42800/3), (342400/3), 9296960/9, 687731200/63, ...];

n=6: [1, 24, 312, 3024, 25680, (1073856/5), (9664704/5), 690265344/35, ...];

n=7: [1, 28, 420, 13832/3, 129248/3, 1905792/5, (156447424/45), (312894848/9), ...]; ...

in which the coefficients in parenthesis are related by

4 = 4*(1); 40 = 5*(8); 504 = 6*(84); 21728/3 = 7*(3104/3); 342400/3 = 8*(42800/3); 9664704/5 = 9*(1073856/5); ...

illustrating: [x^n] A(x)^(4*n) = (n+3) * [x^(n-1)] A(x)^(4*n).

LOGARITHMIC PROPERTY.

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

log(A(x)) = x * (1 - 3*x*A'(x)/A(x)) / (1 - 4*x*A'(x)/A(x));

explicitly,

log(A(x)) = x + x^2 + 6*x^3 + 50*x^4 + 520*x^5 + 6312*x^6 + 86080*x^7 + 1288704*x^8 + 20862720*x^9 + 361454720*x^10 + ... + A300989(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(4*(#A-1))); A[#A] = ((#A+2)*V[#A-1] - V[#A])/(4*(#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-3*x*A')/(A-4*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}

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

CROSSREFS

Cf. A300989, A182962, A300735, A300986, A300990, A300992.

Cf. A300735, A300870, A300590, A296170.

Sequence in context: A340822 A303159 A274387 * A136648 A114337 A317343

Adjacent sequences:  A300985 A300986 A300987 * A300989 A300990 A300991

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 17 2018

STATUS

approved

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Last modified April 15 04:38 EDT 2021. Contains 342975 sequences. (Running on oeis4.)