A368769
a(n) = (n!)^3 * Sum_{k=1..n} 1/(k!)^3.
Original entry on oeis.org
0, 1, 9, 244, 15617, 1952126, 421659217, 144629111432, 74050105053185, 53982526583771866, 53982526583771866001, 71850742883000353647332, 124158083701824611102589697, 272775309892908670592389564310, 748495450346141392105516964466641
Offset: 0
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Table[(n!)^3 Sum[1/(k!)^3,{k,n}],{n,0,20}] (* Harvey P. Dale, May 11 2025 *)
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a(n) = n!^3*sum(k=1, n, 1/k!^3);
A180255
a(n) = n^2 * a(n-1) + n, a(0)=0.
Original entry on oeis.org
0, 1, 6, 57, 916, 22905, 824586, 40404721, 2585902152, 209458074321, 20945807432110, 2534442699285321, 364959748697086236, 61678197529807573897, 12088926715842284483826, 2720008511064514008860865, 696322178832515586268381456, 201237109682597004431562240801
Offset: 0
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FoldList[#2^2*# + #2 &, Range[0, 20]] (* Paolo Xausa, Jun 19 2025 *)
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a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* Bruno Berselli, May 23 2011 */
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a(n)=if(n==0,0,(n)^2*a(n-1)+(n));
for(n=0,12,print1(a(n),", ")); /* show terms */
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
A346410
a(n) = (n!)^2 * Sum_{k=0..n-1} 1 / ((n-k) * k!)^2.
Original entry on oeis.org
0, 1, 5, 22, 152, 2001, 45097, 1527506, 71864928, 4466430513, 353828600029, 34770661312190, 4148422395161464, 590479899466175681, 98824492409739430401, 19209838771051338898234, 4291488438323868507946880, 1091819942877526843993466529, 313819508664449992611846900981
Offset: 0
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Table[(n!)^2 Sum[1/((n - k) k!)^2, {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] BesselI[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A368787
a(n) = (n+1) * (n!)^2 * Sum_{k=1..n} 1/((k+1) * (k!)^2).
Original entry on oeis.org
0, 1, 7, 85, 1701, 51031, 2143303, 120024969, 8641797769, 777761799211, 85553797913211, 11293101324543853, 1761723806628841069, 320633732806449074559, 67333083889354305657391, 16159940133445033357773841, 4395503716297049073314484753
Offset: 0
Showing 1-5 of 5 results.
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