A110106 a(n) is the number of coverings of 1..n by cyclic words of length 3n, such that each value from 1 to n appears precisely twice. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,2,2,...,n,n}. Repeats of words are allowed in a given covering.
1, 6, 3960, 24151680, 577882166400, 38039350155206400, 5605398331566095462400, 1614162682147590619140096000, 824800497779996439355497811968000
Offset: 0
Examples
a(1)=6: {123, 132} {112, 233} {113, 322} {133, 122} {123, 123} {132, 132}.
Programs
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Mathematica
RecurrenceTable[{(40320 + 328752*n + 78732*n^7 + 6561*n^8 + 1816668*n^3 + 1818369*n^4 + 1102248*n^5 + 398034*n^6 + 1063116*n^2) * a[n] + (-161280 - 508608*n - 453600*n^3 - 173340*n^4 - 34992*n^5 - 2916*n^6 - 661104*n^2) * a[n + 1] + (12432 + 20070*n + 12114*n^2 + 3240*n^3 + 324*n^4) * a[n + 2] - 2*a[n + 3] == 0, a[0] == 1, a[1] == 6, a[2] == 3960}, a, {n, 0, 15}] (* Vaclav Kotesovec, Oct 24 2023 *)
Formula
Differential equation satisfied by F(t)=sum a(n) t^(3n)/(3n!) {F(0) = 1, (6*t^2-12*t^5+t^8)*F(t) + (-4*t^6-2+16*t^3)*(d/dt)F(t) + 4*t^4*(d^2/dt^2)F(t)};
recurrence satisfied by a(n): {(40320 + 328752*n + 78732*n^7 + 6561*n^8 + 1816668*n^3 + 1818369*n^4 + 1102248*n^5 + 398034*n^6 + 1063116*n^2)*a(n) + (-161280 - 508608*n - 453600*n^3 - 173340*n^4 - 34992*n^5 - 2916*n^6 - 661104*n^2)*a(n+1) + (12432 + 20070*n + 12114*n^2 + 3240*n^3 + 324*n^4)*a(n+2) - 2*a(n+3), a(1) = 6, a(0) = 1, a(2) = 3960}.
a(n) ~ 2^n * 3^(4*n + 1/2) * n^(4*n) / exp(4*n). - Vaclav Kotesovec, Oct 24 2023
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