A304399 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^4) / A(x) = 0 for n>0.

1, 16, 2200, 1809920, 4241345876, 20919209023760, 185887334702902784, 2699985099706935115520, 59877289873410663776378876, 1926339929784486079047963326480, 86370374435881318779333300624751016, 5225229347181019896500110654738959018752, 415299644168495653846091996394573044842672676

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Example

G.f.: A(x) = 1 + 16*x + 2200*x^2 + 1809920*x^3 + 4241345876*x^4 + 20919209023760*x^5 + 185887334702902784*x^6 + 2699985099706935115520*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n+1)^4)/A(x) begins:
n=0: [1, -15, -1960, -1745560, -4181956116, -20781289862564, ...;
n=1: [1, 0, -2080, -1776080, -4208350776, -20844203397376, ...;
n=2: [1, 65, 0, -1867600, -4327445336, -21121523038728, ...;
n=3: [1, 240, 26600, 0, -4559454036, -21903515092368, ...;
n=4: [1, 609, 183056, 34416384, 0, -23127137438064, ...;
n=5: [1, 1280, 816480, 344268080, 103140231304, 0, ...;
n=6: [1, 2385, 2840840, 2251489240, 1330416079284, 599753730572516, 0, ...; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)^4)/A(x) = 0 for n>0.
RELATED SERIES.
1 - 1/A(x) = 16*x + 1944*x^2 + 1743616*x^3 + 4180212500*x^4 + 20777109650064*x^5 + 185199596154767936*x^6 + 2693946371100901126144*x^7 + ...
The logarithmic derivative of the g.f. A(x) begins
A'(x)/A(x) = 16 + 4144*x + 5328256*x^2 + 16842055888*x^3 + 104239488218896*x^4 + 1113257196684170944*x^5 + 18878740287619671915136*x^6 + ...

Prog

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

Crossrefs

Cf. A304193, A304398.

Sequence in context: A208148 A265593 A345275 * A186857 A196884 A165126

Adjacent sequences: A304396 A304397 A304398 * A304400 A304401 A304402

Keyword

nonn

Author

Paul D. Hanna, May 14 2018

Status

approved