A336242
a(n) = (n!)^2 * Sum_{d|n} (-1)^(d+1) / (d!)^2.
Original entry on oeis.org
1, 3, 37, 431, 14401, 403199, 25401601, 1216454399, 135339724801, 9877056537599, 1593350922240001, 178056522962841599, 38775788043632640001, 5700041141609893478399, 1757631343928533032960001, 327562346808114783805439999, 126513546505547170185216000001
Offset: 1
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Table[(n!)^2 Sum[(-1)^(d + 1)/(d!)^2, {d, Divisors[n]}], {n, 1, 17}]
nmax = 17; CoefficientList[Series[Sum[(1 - BesselJ[0, 2 x^(k/2)]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2 // Rest
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a(n) = n!^2*sumdiv(n, d, (-1)^(d+1)/d!^2); \\ Michel Marcus, Jul 13 2020
A336248
a(n) = (n!)^n * Sum_{k=0..n} (-1)^k / (k!)^n.
Original entry on oeis.org
1, 0, 1, 26, 20481, 774403124, 2173797080953345, 645067515585218711490294, 27280857986486289638369834192338945, 213095986405176211170558965907644717041658073416, 386654453940903446694477049963665295677203885863801760000000001
Offset: 0
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Table[(n!)^n Sum[(-1)^k/(k!)^n, {k, 0, n}], {n, 0, 10}]
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a(n) = (n!)^n * sum(k=0, n, (-1)^k / (k!)^n); \\ Michel Marcus, Jul 14 2020
A340789
a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(k+1) / (k!)^2.
Original entry on oeis.org
0, 1, 3, 28, 447, 11176, 402335, 19714416, 1261722623, 102199532464, 10219953246399, 1236614342814280, 178072465365256319, 30094246646728317912, 5898472342758750310751, 1327156277120718819918976, 339752006942904017899257855, 98188330006499261172885520096
Offset: 0
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Table[n!^2 Sum[(-1)^(k + 1)/k!^2, {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[(1 - BesselJ[0, 2 Sqrt[x]])/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!^2
A346411
a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k) * k!)^2.
Original entry on oeis.org
0, 1, -3, 4, -8, 1, 353, 27224, 1871840, 147012849, 13684928021, 1514370713340, 197964773810648, 30300949591876913, 5380510834911767033, 1098630080602791984784, 255851291397441057781120, 67450889282916741495608737, 19994198644782014829579657837, 6623096362909598587714211804212
Offset: 0
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Table[(n!)^2 Sum[(-1)^k/((n - k) k!)^2, {k, 0, n - 1}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[PolyLog[2, x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A337151
a(n) = (n!)^2 * Sum_{k=0..n} (-1)^(n-k) * (k+1) / ((n-k)!)^2.
Original entry on oeis.org
1, 1, 5, 53, 977, 27649, 1111429, 60147205, 4213400897, 370767834593, 40025019652901, 5199763957426741, 800136077306754385, 143904538461745813153, 29906871652295426507237, 7111902097369951568209349, 1918658066681198636106335489, 582817397769914314847061436225
Offset: 0
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a:= n-> n!^2 * add((-1)^k*(n-k+1)/k!^2, k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Jan 27 2021
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Table[n!^2 Sum[(-1)^(n - k) (k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2
A368852
a(n) = (n!)^3 * Sum_{k=0..n} (-1)^k/(k!)^3.
Original entry on oeis.org
1, 0, 1, 26, 1665, 208124, 44954785, 15419491254, 7894779522049, 5755294271573720, 5755294271573720001, 7660296675464621321330, 13236992655202865643258241, 29081672863480695818238355476, 79800110337391029325246047426145
Offset: 0