A370569
Number of permutations of [n] having no adjacent 2-cycles and no adjacent 4-cycles.
Original entry on oeis.org
1, 1, 1, 4, 18, 97, 607, 4358, 35523, 324356, 3280902, 36427352, 440515699, 5764104507, 81147821501, 1223090709078, 19651920713844, 335323035157947, 6055709997021397, 115397482250691724, 2314064310772997407, 48711753977589111112, 1073990818947724506060
Offset: 0
A370670
Expansion of Sum_{k>=0} k! * ( x/(1+x^3) )^k.
Original entry on oeis.org
1, 1, 2, 6, 23, 116, 702, 4945, 39726, 358596, 3593759, 39596032, 475750740, 6190873441, 86740653730, 1301942638170, 20842037779079, 354469561697988, 6382795892548194, 121310901632237857, 2426864464216669694, 50975856191753357928, 1121692313538562441535
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1+x^3))^k))
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a(n) = sum(k=0, n\3, (-1)^k*(n-3*k)!*binomial(n-2*k-1, k));
A370674
Expansion of Sum_{k>0} k! * x^k/(1+x^k)^(k+1).
Original entry on oeis.org
1, 0, 9, 14, 125, 702, 5047, 40172, 362949, 3628100, 39916811, 478996746, 6227020813, 87178250922, 1307674370745, 20922789524192, 355687428096017, 6402373702119096, 121645100408832019, 2432902008136718030, 51090942171709621965, 1124000727777128678510
Offset: 1
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my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*x^k/(1+x^k)^(k+1)))
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a(n) = sumdiv(n, d, (-1)^(d-1)*(d+n/d-1)!/(d-1)!);