230708

Ruby


Expansion of e.g.f. theta_3(q)^(k/2)

k == 1 (mod 2) のとき、出力してみた。

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end

def A186690(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && (n / i) % 2 == 1}
  (-1) ** (n + 1) * s
end

def B(k, n)
  a = [0] + (1..n).map{|i| A186690(i)}
  a_ary = [1]
  (1..n).each{|i|
    m = f(i - 1)
    a_ary << (1..i).inject(0){|s, j| s + k * a[j] * a_ary[-j] * m / f(i - j)}
  }
  a_ary
end

n = 10
-25.step(25, 2){|i| p [i, B(i, n)]}

出力結果

[-25, [1, -25, 675, -19575, 606225, -19944225, 693835875, -25422933375, 977745218625, -39350424479625, 1652904664495875]]

[-23, [1, -23, 575, -15525, 449673, -13887975, 454991175, -15742117125, 573015200625, -21871632491415, 872889859083375]]

[-21, [1, -21, 483, -12075, 325521, -9396765, 288749475, -9398322675, 322652227905, -11641352627205, 440043353374275]]

[-19, [1, -19, 399, -9177, 228969, -6146595, 176336055, -5376116025, 173372815665, -5890443433875, 210126743128575]]

[-17, [1, -17, 323, -6783, 155601, -3861465, 102864195, -2922828615, 88124295105, -2806906632705, 94095203991075]]

[-15, [1, -15, 255, -4845, 101385, -2309535, 56759175, -1494125325, 41881856625, -1244022735375, 38993819043375]]

[-13, [1, -13, 195, -3315, 62673, -1299285, 29228355, -707636475, 18316505025, -504116658045, 14685387955875]]

[-11, [1, -11, 143, -2145, 36201, -675675, 13777335, -304008705, 7204902705, -182276917515, 4897582264575]]

[-9, [1, -9, 99, -1287, 19089, -316305, 5772195, -114729615, 2462398785, -56674425465, 1390839617475]]

[-7, [1, -7, 63, -693, 8841, -127575, 2047815, -36081045, 690863985, -14263534215, 315469515375]]

[-5, [1, -5, 35, -315, 3345, -40845, 562275, -8575875, 143273025, -2598830325, 50790003075]]

[-3, [1, -3, 15, -105, 873, -8595, 97335, -1233225, 17298225, -266220675, 4444840575]]

[-1, [1, -1, 3, -15, 81, -585, 4995, -46935, 499905, -6109425, 79791075]]

[1, [1, 1, -1, 3, 9, -15, 135, -2205, 21105, 76545, 694575]]

[3, [1, 3, 3, -3, 81, 315, -765, 4725, 16065, 1091475, 10843875]]

[5, [1, 5, 15, 15, 105, 1845, 5175, -11025, 363825, 4011525, 46734975]]

[7, [1, 7, 35, 105, 273, 4095, 38115, 86625, 540225, 22444695, 215417475]]

[9, [1, 9, 63, 315, 1161, 8505, 112455, 796635, 2843505, 57661065, 1099313775]]

[11, [1, 11, 99, 693, 3729, 22275, 259875, 2900205, 19490625, 161008155, 3451399875]]

[13, [1, 13, 143, 1287, 9321, 62205, 598455, 7702695, 78423345, 647284365, 9370936575]]

[15, [1, 15, 195, 2145, 19665, 158535, 1447875, 18243225, 232985025, 2435846175, 28443399075]]

[17, [1, 17, 255, 3315, 36873, 358785, 3490695, 42117075, 590951025, 7589122065, 91863766575]]

[19, [1, 19, 323, 4845, 63441, 731595, 7979715, 96556005, 1388717505, 20396897955, 278955792675]]

[21, [1, 21, 399, 6783, 102249, 1370565, 16991415, 216411615, 3146775345, 49819225365, 767467173375]]

[23, [1, 23, 483, 9177, 156561, 2398095, 33725475, 465686865, 6952917825, 114751084455, 1935277553475]]

[25, [1, 25, 575, 12075, 230025, 3969225, 62850375, 953260875, 14929952625, 254408009625, 4570418955375]]