仕事の部屋: 9月 2024

2024年9月18日水曜日

240918

PARI

(1/x) * Series_Reversion( x * f(x) ).

a(n) = (1/(n+1)) * [x^n] (1/f(x))^(n+1) となることを確認する。

N=20;
x='x+O('x^(N+2));

f = 1/(1 + 5*x + x^2);
a = sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k);
b = serreverse(x*f)/x;
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

f = 1 + 5*x + x^2;
a = sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k);
b = serreverse(x*f)/x;
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

f = 1/(3 - 2*exp(x));
a = serlaplace(sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k));
b = serlaplace(serreverse(x*f)/x);
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

f = 3 - 2*exp(x);
a = serlaplace(sum(k=0, N, polcoef(1/f^(k+1), k)/(k+1) * x^k));
b = serlaplace(serreverse(x*f)/x);
print(Vec(a));
print(Vec(b));
print(Vec(a-b));

出力結果

[1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171, 0]

[]

[1, -5, 49, -600, 8228, -120890, 1860738, -29616675, 483483055, -8050547505, 136201191477, -2334603084060, 40456777529028, -707614530573800, 12475631038479848, -221479557759651840, 3955846194794651388, -71035215358997870750, 1281683885259534013950, -23224445624372453734350, 422457775863392772285090, 0]

[]

[1, -2, 6, -14, -82, 2058, -22778, 55186, 4737630, -141417830, 1940770358, 18532189410, -2031211536242, 63333969461098, -624072759994266, -46503997775007182, 3129486110236404926, -90436659990999596742, -559947921342589721450, 214981434079884273561730, -12114419920158647173081362, 0]

[]

[1, 2, 18, 302, 7562, 253542, 10685794, 543309230, 32378850042, 2214215333750, 170939286647570, 14707184259036414, 1395561779648175274, 144795755972202587462, 16308198003201872476866, 1981633767850818093910094, 258406311809937562215099482, 35994776359231593721760238102, 5334228760604574619939352424562, 837995152675419667859461838034590, 139110781237167610980642890783413770, 0]

[]