A334670
a(n) = (2*n+1)!! * (Sum_{k=1..n} 1/(2*k+1)).
Original entry on oeis.org
0, 1, 8, 71, 744, 9129, 129072, 2071215, 37237680, 741975345, 16236211320, 387182170935, 9995788416600, 277792140828825, 8269430130712800, 262542617405726175, 8855805158351474400, 316285840413064454625, 11924219190760084593000, 473245342972281190686375, 19722890048636406588957000
Offset: 0
a(1) = 3 * (1/3) = 1.
a(2) = 3*5 * (1/3 + 1/5) = 8.
a(3) = 3*5*7 * (1/3 +1/5 + 1/7) = 71.
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a[n_] := (2*n + 1)!! * Sum[1/(2*k + 1), {k, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Apr 29 2021 *)
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{a(n) = prod(k=1, n, 2*k+1)*sum(k=1, n, 1/(2*k+1))}
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{a(n) = if(n<2, n, 4*n*a(n-1)-(2*n-1)^2*a(n-2))}
A335090
a(n) = ((2*n+1)!!)^2 * (Sum_{k=1..n} 1/(2*k+1)^2).
Original entry on oeis.org
0, 1, 34, 1891, 164196, 20760741, 3616621254, 832001250375, 244557191709000, 89472598178279625, 39886085958271670250, 21288783013213520392875, 13405493416599700058947500, 9835107221539462476348118125, 8316889511005794888839427108750, 8030850428074789829954674314399375
Offset: 0
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a[n_] := ((2*n + 1)!!)^2 * Sum[1/(2*k + 1)^2, {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Apr 29 2021 *)
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{a(n) = prod(k=1, n, 2*k+1)^2*sum(k=1, n, 1/(2*k+1)^2)}
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{a(n) = if(n<2, n, (8*n^2+2)*a(n-1)-(2*n-1)^4*a(n-2))}
A335091
a(n) = ((2*n+1)!!)^3 * (Sum_{k=1..n} 1/(2*k+1)^3).
Original entry on oeis.org
0, 1, 152, 55511, 41625144, 56246975289, 124697847089808, 423322997436687375, 2088114588247920714000, 14363296872939657999716625, 133299155158711610547152961000, 1624450039177408057102079622846375, 25413656551949715361011431877529125000, 500711137690193661025654228810320074015625
Offset: 0
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a[n_] := ((2*n + 1)!!)^3 * Sum[1/(2*k + 1)^3, {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Apr 29 2021 *)
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{a(n) = prod(k=1, n, 2*k+1)^3*sum(k=1, n, 1/(2*k+1)^3)}
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{a(n) = if(n<2, n, ((2*n-1)^3+(2*n+1)^3)*a(n-1)-(2*n-1)^6*a(n-2))}
A335092
a(n) = ((2*n+1)!!)^4 * (Sum_{k=1..n} 1/(2*k+1)^4).
Original entry on oeis.org
0, 1, 706, 1745731, 11575291716, 170271339664581, 4874795836698898566, 247120020454614424554375, 20656593715240068513643845000, 2693397991748017956223512587135625, 523998492940635622166679925147692626250
Offset: 0
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a[n_] := ((2*n + 1)!!)^4 * Sum[1/(2*k + 1)^4, {k, 1, n}]; Array[a, 11, 0] (* Amiram Eldar, Apr 28 2021 *)
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{a(n) = prod(k=1, n, 2*k+1)^4*sum(k=1, n, 1/(2*k+1)^4)}
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{a(n) = if(n<2, n, ((2*n-1)^4+(2*n+1)^4)*a(n-1)-(2*n-1)^8*a(n-2))}
A335096
a(n) = ((2*n+1)!!)^n * (Sum_{k=1..n} 1/(2*k+1)^n).
Original entry on oeis.org
0, 1, 34, 55511, 11575291716, 548347875819272649, 8811385079228718926321932614, 66303398534111438173105653188803948359375, 308529654991526005900670429792887300937160115962403125000
Offset: 0
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Table[((2*n + 1)!!)^n * Sum[1/(2*k + 1)^n, {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 25 2020 *)
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{a(n) = prod(k=1, n, 2*k+1)^n*sum(k=1, n, 1/(2*k+1)^n)}
Showing 1-5 of 5 results.