A304398
G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^3) / A(x) = 0 for n>0.
Original entry on oeis.org
1, 8, 199, 19568, 4309702, 1628514128, 927231430126, 737350581437744, 778840734924755140, 1054020790695331268000, 1778132840285207445942196, 3659007006256230147804241040, 9023119928096184018484024831288, 26274442260784898029809836586675872, 89218495222818281880277619804533375624, 349496587851612327547463367678217875791792
Offset: 0
G.f.: A(x) = 1 + 8*x + 199*x^2 + 19568*x^3 + 4309702*x^4 + 1628514128*x^5 + 927231430126*x^6 + 737350581437744*x^7 + 778840734924755140*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n+1)^3)/A(x) begins:
n=0: [1, -7, -143, -17031, -4008021, -1560094653, -901603927833, ...;
n=1: [1, 0, -171, -18144, -4130451, -1588513680, -912609360075, ...;
n=2: [1, 19, 0, -20424, -4500552, -1670248944, -943515644316, ...;
n=3: [1, 56, 1369, 0, -5042565, -1848681000, -1008460310529, ...;
n=4: [1, 117, 6615, 221979, 0, -2071834128, -1129354648380, ...;
n=5: [1, 208, 21357, 1424544, 64174929, 0, -1267137137679, ...;
n=6: [1, 335, 55774, 6134466, 495645999, 29071716177, 0, ...; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)^3)/A(x) = 0 for n>0.
RELATED SERIES.
1 - 1/A(x) = 8*x + 135*x^2 + 16896*x^3 + 3991125*x^4 + 1556103528*x^5 + 900047824305*x^6 + 722051918333952*x^7 + 766786063398540525*x^8 + ...
The logarithmic derivative of the g.f. A(x) begins
A'(x)/A(x) = 8 + 334*x + 54440*x^2 + 16580278*x^3 + 7958081528*x^4 + 5480891617798*x^5 + 5107502440681208*x^6 + 6182250826385760238*x^7 + ...
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{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^3)/Ser(A) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
A304644
G.f. A(x) satisfies: [x^n] (1+x)^(n^4) / A(x) = 0 for n>0.
Original entry on oeis.org
1, 1, 105, 73865, 149937065, 663916103529, 5451834603894529, 74704077908738108545, 1585534054417382287240065, 49309970434271232435701612225, 2152501158830776821954197582557961, 127436616988374904669593064111888541481, 9949767410829299590962659524265243208970825, 1000853528058644375639385529872204384996958065865, 127177120321862418629253989604625620834052796464647105
Offset: 0
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 + ...
-
{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), ", "))
Showing 1-2 of 2 results.