200711

PARI


A336163 とA336179

A268545 のGheorghe Coserea さんのコード参考に

diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z)

と

diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z)

が一致することを確認してみた。

(21:46) gp > N=10;
(21:46) gp > diag(n, expr, var=variables(expr)) = {
  my(a=vector(n));
  for(i=1, #var, expr=taylor(expr, var[#var-i+1], n));
  for(j=1, n, a[j]=expr;
    for(i=1, #var, a[j]=polcoeff(a[j], j-1)));
  return(a);
};
(21:46) gp > R_1(k) = 1/(1+y+z+x*y+y*z+k*z*x+(k+1)*x*y*z);
(21:46) gp > R_2(k) = 1/((1-x)*(1-y)*(1-z)-k*x*y*z);
(21:46) gp > for(k=-10, 10, v_1=diag(N, R_1(k)); v_2=diag(N, R_2(k)); if(v_1==v_2, print(v_1)))
[1, -9, 21, 1431, -33039, 248751, 5485341, -206302329, 2626974081, 17422114311]
[1, -8, 10, 1216, -23174, 111952, 4359664, -118557440, 1014582682, 17402599792]
[1, -7, 1, 1001, -15359, 30233, 3126529, -61392247, 259448833, 11970181433]
[1, -6, -6, 792, -9414, -11556, 2010576, -27431424, -20457702, 6451563924]
[1, -5, -11, 595, -5135, -26525, 1133245, -9654125, -75662975, 2718080875]
[1, -4, -14, 416, -2294, -25624, 533296, -1992064, -52440134, 824947256]
[1, -3, -15, 261, -639, -17523, 188049, 284085, -21320703, 137339037]
[1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372]
[1, -1, -11, 47, 241, -2281, -3779, 104831, -110207, -4415281]
[1, 0, -6, 0, 90, 0, -1680, 0, 34650, 0]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932]
[1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923]
[1, 4, 34, 352, 3946, 46744, 573616, 7217536, 92527738, 1203467464]
[1, 5, 49, 605, 8065, 113525, 1656145, 24774125, 377601025, 5839329125]
[1, 6, 66, 936, 14346, 231876, 3885456, 66767616, 1169068986, 20769386796]
[1, 7, 85, 1351, 23281, 422527, 7951069, 153458935, 3018043777, 60225528727]
[1, 8, 106, 1856, 35386, 709808, 14762224, 314931968, 6846486298, 151025959568]
[1, 9, 129, 2457, 51201, 1121769, 25477761, 593640441, 14095941633, 339631646409]
[1, 10, 154, 3160, 71290, 1690300, 41536720, 1047232000, 26908298650, 701590424500]
[1, 11, 181, 3971, 96241, 2451251, 64689661, 1751654531, 48341416321, 1353802736891]
(21:46) gp >