A361626
Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^2.
Original entry on oeis.org
1, 3, 17, 139, 1437, 17711, 252133, 4059567, 72779129, 1435276027, 30836352441, 716101686323, 17858449006357, 475653606922599, 13467411746316557, 403708230041927191, 12767545998797849073, 424670548932688771187, 14814998283177691422049
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^2))
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a(n)=n! * sum(k=0, n, binomial(n+2*k+1,n-k)/k!) \\ Winston de Greef, Mar 18 2023
A386514
Expansion of e.g.f. exp(x^2/(1-x)^3).
Original entry on oeis.org
1, 0, 2, 18, 156, 1560, 18480, 254520, 3973200, 68947200, 1312748640, 27175024800, 607314818880, 14566195163520, 373027570755840, 10154293067318400, 292659790712889600, 8899747730037964800, 284685195814757337600, 9553060139009702515200, 335468448755976164428800
Offset: 0
a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.
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nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)
A380641
Expansion of e.g.f. exp(x/(1 - 3*x)^3).
Original entry on oeis.org
1, 1, 19, 379, 8857, 244801, 7904251, 292980619, 12257946289, 570627408097, 29212843607011, 1629314013114811, 98250285167099209, 6365331315043185889, 440712959779710869707, 32460639303987670526731, 2533396174719346231613281, 208776665140069914314618689
Offset: 0
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Table[n! * Sum[3^k * Binomial[3*n-2*k-1,k]/(n-k)!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jan 29 2025 *)
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a(n) = n!*sum(k=0, n, 3^k*binomial(3*n-2*k-1, k)/(n-k)!);
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