A304644 - OEIS

%I #5 May 15 2018 16:55:33

%S 1,1,105,73865,149937065,663916103529,5451834603894529,

%T 74704077908738108545,1585534054417382287240065,

%U 49309970434271232435701612225,2152501158830776821954197582557961,127436616988374904669593064111888541481,9949767410829299590962659524265243208970825,1000853528058644375639385529872204384996958065865,127177120321862418629253989604625620834052796464647105

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

%e 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 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in (1+x)^(n^4) / A(x) begins:

%e n=0: [1, -1, -104, -73656, -149778624, -663600972000, -5450470326008280, ...];

%e n=1: [1, 0, -105, -73760, -149852280, -663750750624, -5451133926980280, ...];

%e n=2: [1, 15, 0, -74880, -150968340, -666006324396, -5461105956428160, ...];

%e n=3: [1, 80, 3055, 0, -154503300, -675956601408, -5504713445922300, ...];

%e n=4: [1, 255, 32280, 2630600, 0, -696001081248, -5625102954138200, ...];

%e n=5: [1, 624, 194271, 40161344, 6040383876, 0, -5818032088967780, ...];

%e n=6: [1, 1295, 837760, 360910080, 116308352940, 29159359047060, 0, ...];

%e n=7: [1, 2400, 2878695, 2300795040, 1378317489120, 659313875405856, 255975781942704720, 0, ...]; ...

%e 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.

%e RELATED SERIES.

%e 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 + ...

%e The logarithmic derivative of the g.f. A(x) begins

%e 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 + ...

%o (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]}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A304191, A304643, A304399.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 15 2018