A352660
a(n) = n! * Sum_{k=0..floor(n/4)} 1 / (4*k)!.
Original entry on oeis.org
1, 1, 2, 6, 25, 125, 750, 5250, 42001, 378009, 3780090, 41580990, 498971881, 6486634453, 90812882342, 1362193235130, 21795091762081, 370516559955377, 6669298079196786, 126716663504738934, 2534333270094778681, 53220998671990352301, 1170861970783787750622
Offset: 0
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Table[n! Sum[1/(4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Cos[x] + Cosh[x])/(2 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
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a(n) = n! * sum(k=0, n\4, 1/(4*k)!); \\ Michel Marcus, Mar 29 2022
A348597
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k / (3*k)!.
Original entry on oeis.org
1, 1, 2, 5, 20, 100, 601, 4207, 33656, 302903, 3029030, 33319330, 399831961, 5197815493, 72769416902, 1091541253529, 17464660056464, 296899220959888, 5344185977277985, 101539533568281715, 2030790671365634300, 42646604098678320299, 938225290170923046578
Offset: 0
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Table[n! Sum[(-1)^k/(3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Exp[-x] + 2 Exp[x/2] Cos[Sqrt[3] x/2])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
A349088
a(n) = n! * Sum_{k=0..floor((n-1)/3)} 1 / (3*k+1)!.
Original entry on oeis.org
0, 1, 2, 6, 25, 125, 750, 5251, 42008, 378072, 3780721, 41587931, 499055172, 6487717237, 90828041318, 1362420619770, 21798729916321, 370578408577457, 6670411354394226, 126737815733490295, 2534756314669805900, 53229882608065923900, 1171057417377450325801
Offset: 0
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Table[n! Sum[1/(3 k + 1)!, {k, 0, Floor[(n - 1)/3]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Exp[x] - 2 Exp[-x/2] Sin[(Pi - 3 Sqrt[3] x)/6])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
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