220118

PARI


分割における最小の数と分割の数に関する数列

次のような美しい式が成り立つ。

N=40; x='x+O('x^N);

f_x = sum(k=0, N, x^(k*(k+1)/2)  /prod(j=1, k, 1-x^j));
f_y = sum(k=0, N, x^(k*(3*k-1)/2)/prod(j=1, k, 1-x^j));
f_z = sum(k=0, N, x^(k*(3*k+1)/2)/prod(j=1, k, 1-x^j));

g_x = sum(k=0, N, x^k            /prod(j=1, k, 1-x^j));
g_y = sum(k=0, N, x^k^2          /prod(j=1, k, 1-x^j));
g_z = sum(k=0, N, x^(k*(k+1))    /prod(j=1, k, 1-x^j));

print(Vec(f_x))       \\ A000009
print(Vec(f_y))       \\ A025157
print(Vec(f_z))       \\ A237979
print(Vec(f_y - f_z)) \\ A096401
print(Vec(f_x - f_y)) \\ A237976
print(Vec(f_x - f_z)) \\ A237977

print(Vec(g_x))       \\ A000041
print(Vec(g_y))       \\ A003114
print(Vec(g_z))       \\ A003106
print(Vec(g_y - g_z)) \\ A006141
print(Vec(g_x - g_y)) \\ A039899
print(Vec(g_x - g_z)) \\ A039900

出力結果

[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982]

[1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 66, 73, 82, 91, 102, 113, 126, 139, 155, 171]

[1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 28, 32, 35, 40, 45, 50, 56, 63, 70, 78, 87, 96, 107, 118, 131]

[1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 26, 28, 32, 35, 39, 43, 48, 53]

[1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 14, 17, 21, 25, 31, 37, 45, 54, 64, 76, 90, 106, 124, 146, 170, 198, 230, 267, 308, 357, 410, 472, 542, 621, 709, 811]

[1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 29, 36, 42, 51, 60, 72, 84, 100, 117, 137, 160, 187, 216, 251, 290, 334, 385, 442, 507, 581, 664, 757, 864]

[1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185]

[1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333]

[1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 16, 20, 22, 26, 29, 35, 38, 45, 50, 58, 64, 75, 82, 95, 105, 120, 133, 152, 167, 190, 210, 237]

[1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123]

[1, 2, 3, 5, 8, 12, 18, 25, 36, 49, 68, 91, 123, 162, 214, 278, 362, 464, 596, 757, 961, 1209, 1521, 1897, 2366, 2931, 3627, 4463, 5487, 6711, 8200, 9976, 12121, 14672, 17738, 21371, 25716, 30852]

[1, 1, 2, 4, 6, 9, 13, 19, 27, 38, 52, 71, 95, 127, 167, 220, 285, 370, 474, 607, 770, 976, 1226, 1540, 1920, 2391, 2960, 3660, 4501, 5529, 6760, 8254, 10038, 12190, 14750, 17825, 21470, 25825, 30975]