A337152
a(n) = 2^n * (n!)^2 * Sum_{k=0..n} 1 / ((-2)^k * (k!)^2).
Original entry on oeis.org
1, 1, 9, 161, 5153, 257649, 18550729, 1817971441, 232700344449, 37697455800737, 7539491160147401, 1824556860755671041, 525472375897633259809, 177609663053400041815441, 69622987916932816391652873, 31330344562619767376243792849, 16041136416061320896636821938689
Offset: 0
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Table[2^n n!^2 Sum[1/((-2)^k k!^2), {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!^2
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a(n) = 2^n * (n!)^2 * sum(k=0, n, 1 / ((-2)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021
A337154
a(n) = 4^n * (n!)^2 * Sum_{k=0..n} 1 / ((-4)^k * (k!)^2).
Original entry on oeis.org
1, 3, 49, 1763, 112833, 11283299, 1624795057, 318459831171, 81525716779777, 26414332236647747, 10565732894659098801, 5113814721015003819683, 2945557279304642200137409, 1991196720809938127292888483, 1561098229114991491797624570673, 1404988406203492342617862113605699
Offset: 0
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Table[4^n n!^2 Sum[1/((-4)^k k!^2), {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 4 x), {x, 0, nmax}], x] Range[0, nmax]!^2
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a(n) = 4^n * (n!)^2 * sum(k=0, n, 1 / ((-4)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021
A337155
a(n) = 5^n * (n!)^2 * Sum_{k=0..n} 1 / ((-5)^k * (k!)^2).
Original entry on oeis.org
1, 4, 81, 3644, 291521, 36440124, 6559222321, 1607009468644, 514243029966081, 208268427136262804, 104134213568131402001, 63001199208719498210604, 45360863430278038711634881, 38329929598584942711331474444, 37563331006613243857104844955121, 42258747382439899339242950574511124
Offset: 0
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Table[5^n n!^2 Sum[1/((-5)^k k!^2), {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 5 x), {x, 0, nmax}], x] Range[0, nmax]!^2
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a(n) = 5^n * (n!)^2 * sum(k=0, n, 1 / ((-5)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021
Showing 1-3 of 3 results.