A335095
Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n+1)!!)^k * Sum_{j=1..n} 1/(2*j+1)^k.
Original entry on oeis.org
0, 0, 1, 0, 1, 2, 0, 1, 8, 3, 0, 1, 34, 71, 4, 0, 1, 152, 1891, 744, 5, 0, 1, 706, 55511, 164196, 9129, 6, 0, 1, 3368, 1745731, 41625144, 20760741, 129072, 7, 0, 1, 16354, 57365351, 11575291716, 56246975289, 3616621254, 2071215, 8
Offset: 0
Square array begins:
0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, ...
2, 8, 34, 152, 706, ...
3, 71, 1891, 55511, 1745731, ...
4, 744, 164196, 41625144, 11575291716, ...
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T[n_, k_] := ((2*n + 1)!!)^k * Sum[1/(2*j + 1)^k, {j, 1, n}]; Table[T[k, n - k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)
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{T(n, k) = prod(j=1, n, 2*j+1)^k*sum(j=1, n, 1/(2*j+1)^k)}
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))}
Showing 1-4 of 4 results.