求 (1+x+1／x)^100 展开式的常数项

中国上海市

求（1+x+1/x）∧100的常数项

lihp2020

一猜都是很大的数（2^100，3^100）之间  大概在2.8^100 左右  所以 应该不能用一个值表示 验证了
0        1
1        1
2        3
3        7
4        19
5        51
6        141
7        393
8        1107
9        3139
10        8953
11        25653
12        73789
13        212971
14        616227
15        1787607
这个前15项 没有发现规律   后面想到了  王守恩   给我的oeis  在上面搜索 刚好出来这个A002426

由于是英文的我好想有点看不懂  一定是这个序列 好想后面提示有解法 但是 读不懂

天山草

展开式的常数项为  25134265191388162956642519120384003897467908119。

luyuanhong

luyuanhong

王守恩

{1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607,
5196627, 15134931, 44152809, 128996853, 377379369, 1105350729,  3241135527,
9513228123,  27948336381,  82176836301,  241813226151,  712070156203, ...........}

\(\displaystyle a_{n}=\frac{(2n-1)a_{(n-1)}+3(n-1)a_{(n-2)}}{n}\)

\(\displaystyle a_{n}=\sum_{k=0}^{n/3}\frac{n(2n-3k-1)!\cos(n\pi)}{k!(n-k)!(n-3k)!}\)

\(\displaystyle a_{n}=\sum_{k=0}^{n/2}\frac{n!}{(k!)^2(n-2k)!}\)

王守恩

王守恩 发表于 2021-10-13 18:47
{1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607,
5196627, 1513 ...

Table[Table[CoefficientList[Series[(1 + x + x^2)^n, {x, 0, n}], x][[a]], {n, a - 1, 23}], {a, 1, 17}]
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23},
{3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91,105, 120, 136, 153, 171, 190, 210, 231, 253, 276},
{7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002,},
{19, 45, 90, 161, 266, 414, 615, 880, 1221,1651, 2184, 2835, 3620, 4556, 5661, 6954, 8455, 10185,},
{51, 126, 266, 504, 882, 1452, 2277, 3432, 5005, 7098, 9828,13328, 17748, 23256, 30039, 38304,},
{141, 357, 784, 1554, 2850, 4917, 8074, 12727, 19383, 28665, 41328, 58276, 80580, 109497, 146490,},
{393, 1016, 2304, 4740, 9042, 16236, 27742, 45474, 71955, 110448, 165104, 241128, 344964, 484500,},
{1107, 2907, 6765, 14355, 28314, 52624, 93093, 157950, 258570, 410346, 633726, 955434, 1409895,},
{3139, 8350, 19855, 43252, 87802, 168168, 306735, 536640, 905658, 1481108, 2355962, 3656360,},
{8953, 24068, 58278, 129844, 270270,  531531, 996216, 1791426, 3107430, 5222264, 8533660,},
{25653, 69576, 171106, 388752, 827190, 1665456, 3198312, 5895396, 10483934, 18062160,},
{73789, 201643, 502593, 1161615, 2520336, 5182008, 10171746, 19174572, 34880770, 61476590,},
{212941, 585690, 1477035, 3465840, 7651632, 16031952, 32099094, 61757600, 114700530,},
{616227, 1704510, 4343160, 10329336, 23162976, 49366674, 100640340, 197278710, 373455830,},

王守恩

王守恩 发表于 2021-10-13 18:47
{1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607,
5196627, 1513 ...

数！是相通的(这是"杨辉三角")!
Table[Table[CoefficientList[Series[(1 + x)^n, {x, 1, n}], x][[a + 1]], {n, a, 23}], {a, 0, 17}]
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,19, 20, 21, 22, 23},
{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253},
{1, 4, 10, 20, 35,56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771},
{1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001,1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855},
{1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628,  15504, 20349, 26334,},
{1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613,},
{1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544,},
{1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314},
{1, 10, 55, 220, 715, 2002, 5005, 11440, 24310,  48620, 92378, 167960, 293930, 497420, 817190},
{1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646,  1144066},
{1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078},
{1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078},
{1, 14, 105, 560, 2380, 8568,  27132, 77520, 203490, 497420, 1144066},
{1, 15, 120, 680, 3060,  11628, 38760, 116280, 319770, 817190},
{1, 16, 136, 816, 3876,  15504, 54264, 170544, 490314},
{1, 17, 153, 969, 4845, 20349, 74613, 245157},
{1, 18, 171, 1140, 5985, 26334, 100947}}