prd=1
for n=1:50
prd=prd*(4-2/sym(n));
disp([n prd])
end
produces
n product
[1, 2]
[2, 6]
[3, 20]
[4, 70]
[5, 252]
[6, 924]
[7, 3432]
[8, 12870]
[9, 48620]
[10, 184756]
[11, 705432]
[12, 2704156]
[13, 10400600]
[14, 40116600]
[15, 155117520]
[16, 601080390]
[17, 2333606220]
[18, 9075135300]
[19, 35345263800]
[20, 137846528820]
[21, 538257874440]
[22, 2104098963720]
[23, 8233430727600]
[24, 32247603683100]
[25, 126410606437752]
[26, 495918532948104]
[27, 1946939425648112]
[28, 7648690600760440]
[29, 30067266499541040]
[30, 118264581564861424]
[31, 465428353255261088]
[32, 1832624140942590534]
[33, 7219428434016265740]
[34, 28453041475240576740]
[35, 112186277816662845432]
[36, 442512540276836779204]
[37, 1746130564335626209832]
[38, 6892620648693261354600]
[39, 27217014869199032015600]
[40, 107507208733336176461620]
[41, 424784580848791721628840]
[42, 1678910486211891090247320]
[43, 6637553085023755473070800]
[44, 26248505381684851188961800]
[45, 103827421287553411369671120]
[46, 410795449442059149332177040]
[47, 1625701140345170250548615520]
[48, 6435067013866298908421603100]
[49, 25477612258980856902730428600]
[50, 100891344545564193334812497256]
Not only is each of these first fifty an integer, but it also seems to be OEIS's
A000984: Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2, which are definitely all integers.
so
prd=1
for n=1:50
prd=prd*(4-2/sym(n));
disp([n prd])
disp(factorial(2*sym(n))/factorial(sym(n))^2)
end
verifies the identification
[1, 2]
2
[2, 6]
6
[3, 20]
20
[4, 70]
70
[5, 252]
252
[6, 924]
924
[7, 3432]
3432
[8, 12870]
12870
[9, 48620]
48620
[10, 184756]
184756
[11, 705432]
705432
[12, 2704156]
2704156
[13, 10400600]
10400600
[14, 40116600]
40116600
[15, 155117520]
155117520
[16, 601080390]
601080390
[17, 2333606220]
2333606220
[18, 9075135300]
9075135300
[19, 35345263800]
35345263800
[20, 137846528820]
137846528820
[21, 538257874440]
538257874440
[22, 2104098963720]
2104098963720
[23, 8233430727600]
8233430727600
[24, 32247603683100]
32247603683100
[25, 126410606437752]
126410606437752
[26, 495918532948104]
495918532948104
[27, 1946939425648112]
1946939425648112
[28, 7648690600760440]
7648690600760440
[29, 30067266499541040]
30067266499541040
[30, 118264581564861424]
118264581564861424
[31, 465428353255261088]
465428353255261088
[32, 1832624140942590534]
1832624140942590534
[33, 7219428434016265740]
7219428434016265740
[34, 28453041475240576740]
28453041475240576740
[35, 112186277816662845432]
112186277816662845432
[36, 442512540276836779204]
442512540276836779204
[37, 1746130564335626209832]
1746130564335626209832
[38, 6892620648693261354600]
6892620648693261354600
[39, 27217014869199032015600]
27217014869199032015600
[40, 107507208733336176461620]
107507208733336176461620
[41, 424784580848791721628840]
424784580848791721628840
[42, 1678910486211891090247320]
1678910486211891090247320
[43, 6637553085023755473070800]
6637553085023755473070800
[44, 26248505381684851188961800]
26248505381684851188961800
[45, 103827421287553411369671120]
103827421287553411369671120
[46, 410795449442059149332177040]
410795449442059149332177040
[47, 1625701140345170250548615520]
1625701140345170250548615520
[48, 6435067013866298908421603100]
6435067013866298908421603100
[49, 25477612258980856902730428600]
25477612258980856902730428600
[50, 100891344545564193334812497256]
100891344545564193334812497256
|
|
Posted by Charlie
on 2022-05-26 10:43:32 |
verification as integers
