A300596 E.g.f. A(x) satisfies: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4) for n>=1.

1, 1, 17, 13171, 56479849, 738706542221, 22885801082965201, 1448479282286023114807, 169382934361790242266135761, 33954915787325983176711221469529, 10997512067125948734754888814957997361, 5482894935903399886164748355296587003210971, 4041251688669102134446309448401146782811371078137

Offset

0,3

Comments

Compare e.g.f. to: [x^n] exp(x)^(n^4) = n^3 * [x^(n-1)] exp(x)^(n^4) for n>=1.

Links

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

Formula

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300597(n)*x^n, a power series in x with integer coefficients.

Example

E.g.f.: A(x) = 1 + x + 17*x^2/2! + 13171*x^3/3! + 56479849*x^4/4! + 738706542221*x^5/5! + 22885801082965201*x^6/6! + 1448479282286023114807*x^7/7! + 169382934361790242266135761*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^(n^4) begins:
n=1: [(1), (1), 17/2, 13171/6, 56479849/24, 738706542221/120, ...];
n=2: [1, (16), (256), 113168/3, 114614528/3, 1486010366512/15, ...];
n=3: [1, 81, (7857/2), (636417/2), 1671341283/8, 20586397669407/40, ...];
n=4: [1, 256, 34816, (11641088/3), (2980118528/3), 26464517792512/15, ...];
n=5: [1, 625, 400625/2, 271091875/6, (232095075625/24), (145059422265625/24), ...];
n=6: [1, 1296, 850176, 379068336, 133027474176, (243163666719504/5), (315140112068477184/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 256 = 2^4*16; 636417/2 = 3^4*7857/2; 2980118528/3 = 4^4*11641088/3; ...
illustrating that: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + 31779658925496*x^6 + 287364845865893467*x^7 + 4200677982722915635200*x^8 + ... + A300597(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = ((#A-1)^4*V[#A-1] - V[#A])/(#A-1)^4 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A182962, A296170, A300590, A300592, A300594, A300597.

Sequence in context: A257044 A114924 A362066 * A308962 A191964 A230638

Adjacent sequences: A300593 A300594 A300595 * A300597 A300598 A300599

Keyword

nonn

Author

Paul D. Hanna, Mar 09 2018

Status

approved