A304644 G.f. A(x) satisfies: [x^n] (1+x)^(n^4) / A(x) = 0 for n>0.

1, 1, 105, 73865, 149937065, 663916103529, 5451834603894529, 74704077908738108545, 1585534054417382287240065, 49309970434271232435701612225, 2152501158830776821954197582557961, 127436616988374904669593064111888541481, 9949767410829299590962659524265243208970825, 1000853528058644375639385529872204384996958065865, 127177120321862418629253989604625620834052796464647105

Offset

0,3

Links

Table of n, a(n) for n=0..14.

Example

G.f.: A(x) = 1 + x + 105*x^2 + 73865*x^3 + 149937065*x^4 + 663916103529*x^5 + 5451834603894529*x^6 + 74704077908738108545*x^7 + 1585534054417382287240065*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n^4) / A(x) begins:
n=0: [1, -1, -104, -73656, -149778624, -663600972000, -5450470326008280, ...];
n=1: [1, 0, -105, -73760, -149852280, -663750750624, -5451133926980280, ...];
n=2: [1, 15, 0, -74880, -150968340, -666006324396, -5461105956428160, ...];
n=3: [1, 80, 3055, 0, -154503300, -675956601408, -5504713445922300, ...];
n=4: [1, 255, 32280, 2630600, 0, -696001081248, -5625102954138200, ...];
n=5: [1, 624, 194271, 40161344, 6040383876, 0, -5818032088967780, ...];
n=6: [1, 1295, 837760, 360910080, 116308352940, 29159359047060, 0, ...];
n=7: [1, 2400, 2878695, 2300795040, 1378317489120, 659313875405856, 255975781942704720, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n^4) / A(x) = 0 for n>0.
RELATED SERIES.
1 - 1/A(x) = x + 104*x^2 + 73656*x^3 + 149778624*x^4 + 663600972000*x^5 + 5450470326008280*x^6 + 74693014771268857320*x^7 + 1585383397658861643763200*x^8 + ...
The logarithmic derivative of the g.f. A(x) begins
A'(x)/A(x) = 1 + 209*x + 221281*x^2 + 599431169*x^3 + 3318792477121*x^4 + 32706914292746129*x^5 + 522889821925387405441*x^6 + 12683669785848215443184129*x^7 + ...

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-1)^4)/Ser(A) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A304191, A304643, A304399.

Sequence in context: A028917 A145627 A305146 * A171116 A199100 A225217

Adjacent sequences: A304641 A304642 A304643 * A304645 A304646 A304647

Keyword

nonn

Author

Paul D. Hanna, May 15 2018

Status

approved