A288780
Zero together with the row sums of A288778.
Original entry on oeis.org
0, 0, 2, 9, 36, 165, 918, 6111, 47304, 416097, 4091130, 44417043, 527456556, 6798432069, 94499679582, 1408924024695, 22425642181008, 379514672913321, 6804212771165634, 128827325000617947, 2568509718703606260, 53787877376348226573, 1180349932648067726886
Offset: 0
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a:= proc(n) option remember; `if`(n<3, n*(n-1),
n*(a(n-1)*n/(n-1)-a(n-2)*(n-1)/(n-2)))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jun 16 2017
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{0}~Join~Map[Total, Table[(n - k + 1) k! - (k - 1)!, {n, 22}, {k, n}]] (* Michael De Vlieger, Jun 21 2017 *)
A288528
Numbers with consecutive positive decimal digits after the digits are sorted.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 123, 132, 213, 231, 234, 243, 312, 321, 324, 342, 345, 354, 423, 432, 435, 453, 456, 465, 534, 543, 546, 564, 567, 576, 645, 654, 657, 675, 678, 687, 756, 765, 768, 786, 789, 798, 867, 876, 879, 897, 978, 987
Offset: 1
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def ok(n): return "".join(sorted(str(n))) in "123456789"
print([k for k in range(999) if ok(k)]) # Michael S. Branicky, Aug 04 2022
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# alternate for generating full sequence instantly
from itertools import permutations
frags = ["123456789"[i:j] for i in range(9) for j in range(i+1, 10)]
afull = sorted(int("".join(s)) for f in frags for s in permutations(f))
print(afull[:70]) # Michael S. Branicky, Aug 04 2022
A288778
Triangle read by rows (1<=k<=n): T(n,k) = (n-k+1)*k! - (k-1)!
Original entry on oeis.org
0, 1, 1, 2, 3, 4, 3, 5, 10, 18, 4, 7, 16, 42, 96, 5, 9, 22, 66, 216, 600, 6, 11, 28, 90, 336, 1320, 4320, 7, 13, 34, 114, 456, 2040, 9360, 35280, 8, 15, 40, 138, 576, 2760, 14400, 75600, 322560, 9, 17, 46, 162, 696, 3480, 19440, 115920, 685440, 3265920, 10, 19, 52, 186, 816, 4200, 24480, 156240, 1048320, 6894720, 36288000
Offset: 1
Triangle begins:
0;
1, 1;
2, 3, 4;
3, 5, 10, 18;
4, 7, 16, 42, 96;
5, 9, 22, 66, 216, 600;
6, 11, 28, 90, 336, 1320, 4320;
7, 13, 34, 114, 456, 2040, 9360, 35280;
8, 15, 40, 138, 576, 2760, 14400, 75600, 322560;
9, 17, 46, 162, 696, 3480, 19440, 115920, 685440, 3265920;
10, 19, 52, 186, 816, 4200, 24480, 156240, 1048320, 6894720, 36288000;
...
For n = 10 and k = 2; T(10,2) = 17 coincides with the number of positive terms with two digits in A215014 (see the first comment above).
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Table[(n - k + 1) k! - (k - 1)!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 16 2017 *)
Showing 1-3 of 3 results.
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