A345954
a(n) is the number of ternary strings of length n with at least three 0's.
Original entry on oeis.org
0, 0, 0, 1, 9, 51, 233, 939, 3489, 12259, 41385, 135675, 435185, 1373139, 4279161, 13210219, 40490817, 123438531, 374772041, 1134343131, 3425442705, 10326135475, 31088506905, 93507741771, 281053804769, 844319042211, 2535473709033, 7611873722299, 22847398772529, 68567563468179
Offset: 0
a(5)=51 since the strings are the 10 permutations of 11000, the 10 permutations of 22000, the 20 permutations of 12000, the 5 permutations of 10000, the 5 permutations of 20000, and 00000.
Original entry on oeis.org
1, 2, 5, 4, 10, 19, 8, 20, 38, 65, 16, 40, 76, 130, 211, 32, 80, 152, 260, 422, 665, 64, 160, 304, 520, 844, 1330, 2059, 128, 320, 608, 1040, 1688, 2660, 4118, 6305, 256, 640, 1216, 2080, 3376, 5320, 8236, 12610, 19171
Offset: 0
The start of the sequence as a triangle read by rows:
1;
2, 5;
4, 10, 19;
8, 20, 38, 65;
16, 40, 76, 130, 211;
32, 80, 152, 260, 422, 665;
...
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T[n_,k_]:=Sum[3^j*2^(n-j),{j,0,k}];Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Dec 20 2023 *)
A272098
Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.
Original entry on oeis.org
1, 2, 0, 4, 1, 0, 8, 7, 1, 0, 16, 33, 15, 1, 0, 32, 131, 131, 31, 1, 0, 64, 473, 883, 473, 63, 1, 0, 128, 1611, 5111, 5111, 1611, 127, 1, 0, 256, 5281, 26799, 44929, 26799, 5281, 255, 1, 0, 512, 16867, 131275, 344551, 344551, 131275, 16867, 511, 1, 0
Offset: 0
Triangle starts:
[1]
[2, 0]
[4, 1, 0]
[8, 7, 1, 0]
[16, 33, 15, 1, 0]
[32, 131, 131, 31, 1, 0]
[64, 473, 883, 473, 63, 1, 0]
[128, 1611, 5111, 5111, 1611, 127, 1, 0]
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T := (n, k) -> add((-1)^(n-j)*combinat:-eulerian1(j,k)*binomial(-j-1,-n-1), j=0..n): seq(seq(T(n, k), k=0..n), n=0..10);
# Or:
egf := (exp(x)*(y - 1))/(y - exp(x*(y - 1))); ser := series(egf, x, 12):
cx := n -> series(coeff(ser, x, n), y, n + 2):
seq(seq(n!*coeff(cx(n), y, k), k = 0..n), n = 0..9); # Peter Luschny, Aug 14 2022
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<Emanuele Munarini, Oct 19 2023 *)
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