A354298
a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.
Original entry on oeis.org
1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661
Offset: 1
1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...
Cf.
A001147,
A053557,
A061354,
A064646,
A103816,
A113012,
A120265,
A143382,
A289381,
A306858,
A354299 (denominators).
-
S:= 0: R:= NULL:
for n from 1 to 100 do
S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024
-
Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator
A354299
a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.
Original entry on oeis.org
1, 3, 15, 105, 189, 10395, 135135, 2027025, 34459425, 130945815, 13749310575, 316234143225, 7905853580625, 12556355686875, 1238056670725875, 776918153694375, 6332659870762850625, 7642865361265509375, 8200794532637891559375, 63966197354575554163125, 13113070457687988603440625
Offset: 1
1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...
-
S:= 0: R:= NULL:
for n from 1 to 100 do
S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
R:= R, denom(S);
od:
R; # Robert Israel, Jan 10 2024
-
Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Denominator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Denominator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Denominator
A334578
Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.
Original entry on oeis.org
1, 1, 1, 2, 5, 11, 29, 76, 233, 685, 2329, 7534, 27949, 97943, 391285, 1469144, 6260561, 24975449, 112690097, 474533530, 2253801941, 9965204131, 49583642701, 229199695012, 1190007424825, 5729992375301, 30940193045449, 154709794133126, 866325405272573
Offset: 0
a(5) = (5*3*1)*(1/(1) - 1/(3*1) + 1/(5*3*1)) = 15-5+1 = 11.
-
a:= proc(n) option remember; `if`(n<2, [0$2, 1$2][n+3],
(n-1)*a(n-2)+(n-2)*a(n-4))
end:
seq(a(n), n=0..32); # Alois P. Heinz, May 06 2020
-
RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n a[n-2] + (-1)^Floor[n/2]}, a, {n, 0, 32}] (* Jean-François Alcover, Nov 27 2020 *)
A368794
a(n) = (2*n-1)!! * Sum_{k=1..n} (-1)^(k-1)/(2*k-1)!!.
Original entry on oeis.org
0, 1, 2, 11, 76, 685, 7534, 97943, 1469144, 24975449, 474533530, 9965204131, 229199695012, 5729992375301, 154709794133126, 4486584029860655, 139084104925680304, 4589775462547450033, 160642141189160751154, 5943759223998947792699, 231806609735958963915260
Offset: 0
-
a001147(n) = prod(k=1, n, 2*k-1);
a(n) = a001147(n)*sum(k=1, n, (-1)^(k-1)/a001147(k));
Showing 1-4 of 4 results.