A163476
a(n) = 20*a(n-1) - 97*a(n-2) for n > 1; a(0) = 3, a(1) = 33.
Original entry on oeis.org
3, 33, 369, 4179, 47787, 550377, 6372201, 74057451, 863045523, 10077337713, 117831338529, 1379125012419, 16152860411067, 189282082016697, 2218814180460441, 26015921653589211, 305093457567121443
Offset: 0
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[ n le 2 select 30*n-27 else 20*Self(n-1)-97*Self(n-2): n in [1..17] ];
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LinearRecurrence[{20, -97}, {3, 33}, 50] (* G. C. Greubel, Jul 26 2017 *)
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x='x+O('x^50); Vec((3-27*x)/(1-20*x+97*x^2)) \\ G. C. Greubel, Jul 26 2017
A381930
Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1,2} such that I(x) + W_0(x)*W_1(x) + W_0(x)*W_2(x) + W_1(x)*W_2(x) = k where I(x) is the number of inversions in x and W_i(x) is the number of occurrences of the letter i in x for i={0,1,2}, n>=0, 0<=k<=floor(2n^2/3).
Original entry on oeis.org
1, 3, 3, 3, 3, 3, 0, 6, 7, 8, 2, 1, 3, 0, 0, 6, 9, 12, 18, 12, 12, 6, 3, 3, 0, 0, 0, 6, 6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3, 3, 0, 0, 0, 0, 6, 6, 6, 12, 18, 27, 33, 52, 62, 77, 82, 86, 75, 68, 48, 35, 19, 11, 2, 1
Offset: 0
Triangle T(n,k) begins:
1;
3;
3, 3, 3;
3, 0, 6, 7, 8, 2, 1;
3, 0, 0, 6, 9, 12, 18, 12, 12, 6, 3;
3, 0, 0, 0, 6, 6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3;
...
T(3,3) = 7 because we have: {0, 1, 0}, {0, 1, 2}, {0, 2, 0}, {1, 0, 1}, {1, 2, 1}, {2, 0, 2}, {2, 1, 2}.
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b:= proc(n, i, j, k) option remember; expand(
`if`(n=0, z^(i*j+i*k+j*k), b(n-1, i+1, j, k)*z^(j+k)+
b(n-1, i, j+1, k)*z^k +b(n-1, i, j, k+1)))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=0..10); # Alois P. Heinz, Mar 10 2025
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nn = 6; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, q] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^3, {z, 0, nn}],z]] // Grid
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