A010087 Weight distribution of d=3 Hamming code of length 63.

1, 0, 0, 651, 9765, 109368, 1057224, 8649279, 60544953, 369776680, 1996794072, 9621890019, 41694856749, 163568562192, 584173436400, 1908310936455, 5724932809365, 15827726179440, 40448633569680

Offset

0,4

References

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 129.

Links

Georg Fischer, Table of n, a(n) for n = 0..63

M. Terada, J. Asatani and T. Koumoto, Weight Distribution

List of weight distributions

Formula

Recurrence: a(n) = (binomial(m,n-1) - a(n-1) - (m-n+2)*a(n-2))/n for n > 1, a(0)=1, a(1)=0 with m = 63. - Georg Fischer, Apr 14 2020

Example

The weight distribution is:
i A_i
0 1
3 651
4 9765
5 109368
6 1057224
7 8649279
8 60544953
9 369776680
10 1996794072
11 9621890019
12 41694856749
13 163568562192
14 584173436400
15 1908310936455
16 5724932809365
17 15827726179440
18 40448633569680
19 95799462143175
20 210758816714985
21 431553634502760
22 823875120414360
23 1468647185710635
24 2447745309517725
25 3818482327223928
26 5580858785942664
27 7647844002734159
28 9832942289229633
29 11867343566087520
30 13449656041565856
31 14317376396958243
32 14317376396958243
33 13449656041565856
34 11867343566087520
35 9832942289229633
36 7647844002734159
37 5580858785942664
38 3818482327223928
39 2447745309517725
40 1468647185710635
41 823875120414360
42 431553634502760
43 210758816714985
44 95799462143175
45 40448633569680
46 15827726179440
47 5724932809365
48 1908310936455
49 584173436400
50 163568562192
51 41694856749
52 9621890019
53 1996794072
54 369776680
55 60544953
56 8649279
57 1057224
58 109368
59 9765
60 651
63 1

Mathematica

m:=63; RecurrenceTable[{a[n]==(Binomial[m, n-1]-a[n-1]-(m-n+2)*a[n-2])/n,
a[0]==1, a[1]==0}, a, {n, 0, m}] (* _Georg Fischer, Apr 14 2020 *)

Prog

(PARI) Vecrev((1+x)^63 + 63*(1-x)*(1-x^2)^31)/64 \\ Andrew Howroyd, Jan 11 2021

Crossrefs

Row 6 of A340030.

Sequence in context: A255619 A151736 A346138 * A110850 A257715 A048915

Adjacent sequences: A010084 A010085 A010086 * A010088 A010089 A010090

Keyword

nonn,fini,full

Author

N. J. A. Sloane

Status

approved