A247646
Irregular triangle read by rows, arising in enumeration of lattice paths in strips.
Original entry on oeis.org
1, 1, 3, 3, 1, 1, 3, 9, 5, 7, 11, 1, 3, 9, 15, 12, 10, 9, 3, 1, 1, 1, 3, 9, 15, 27, 16, 20, 12, 14, 3, 3, 1, 1
Offset: 0
Triangle begins:
1,
1,3,3,1,
1,3,9,5,7,11,
1,3,9,15,12,10,9,3,1,1,
1,3,9,15,27,16,20,12,14,3,3,1,1,
...
The diagonals on the left are given by
A247643.
A355011
Array read by ascending antidiagonals: T(n, k) is the number of self-conjugate n-core partitions with k corners.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 4, 5, 1, 1, 3, 9, 5, 7, 1, 1, 4, 9, 15, 7, 8, 1, 1, 4, 16, 15, 27, 8, 10, 1, 1, 5, 16, 34, 27, 37, 10, 11, 1, 1, 5, 25, 34, 76, 37, 55, 11, 13, 1, 1, 6, 25, 65, 76, 124, 55, 69, 13, 14, 1, 1, 6, 36, 65, 175, 124, 216, 69, 93, 14, 16, 1, 1
Offset: 2
The array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 5, 7, 8, 10, 11, 13, ...
2, 4, 5, 7, 8, 10, 11, 13, ...
3, 9, 15, 27, 37, 55, 69, 93, ...
3, 9, 15, 27, 37, 55, 69, 93, ...
4, 16, 34, 76, 124, 216, 309, 471, ...
4, 16, 34, 76, 124, 216, 309, 471, ...
5, 25, 65, 175, 335, 675, 1095, 1875, ...
5, 25, 65, 175, 335, 675, 1095, 1875, ...
...
-
T[n_,k_]:=Sum[Binomial[Floor[(k-1)/2],Floor[(i-1)/2]]Binomial[Floor[k/2],Floor[i/2]]Binomial[Floor[n/2]+k-i,k],{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,13},{k,1,n-1}]]
A378809
Triangle read by rows: T(n,k) is the number of peak and valleyless Motzkin meanders of length n with k horizontal steps.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 9, 4, 1, 1, 7, 15, 16, 5, 1, 1, 8, 27, 34, 25, 6, 1, 1, 10, 37, 76, 65, 36, 7, 1, 1, 11, 55, 124, 175, 111, 49, 8, 1, 1, 13, 69, 216, 335, 351, 175, 64, 9, 1, 1, 14, 93, 309, 675, 776, 637, 260, 81, 10, 1
Offset: 0
The triangle begins
k=0 1 2 3 4 5 6 7
n=0 1;
n=1 1, 1;
n=2 1, 2, 1;
n=3 1, 4, 3, 1;
n=4 1, 5, 9, 4, 1;
n=5 1, 7, 15, 16, 5, 1;
n=6 1, 8, 27, 34, 25, 6, 1;
n=7 1, 10, 37, 76, 65, 36, 7, 1;
...
T(3,0) = 1: UUU.
T(3,1) = 4: UUH, UHU, UHD, HUU.
T(3,2) = 3: UHH, HHU, HUH.
T(3,3) = 1: HHH.
-
A088855(n,k) = {binomial(floor((n-1)/2), floor((k-1)/2))*binomial(ceil((n-1)/2),ceil((k-1)/2))}
A_xy(N) = {my(x='x+O('x^N), h = sum(n=0,N, (1/(1-y*x)^(n+1)) * (if(n<1,1,0) + sum(k=1,n, A088855(n,k)*x^(n+k-1)*(y^(k-1)) )) )); for(n=0,N-1,print(Vecrev(polcoeff(h,n))))}
A_xy(10)
A287351
Numbers k such that (10^k*361 - 163)/9 is prime (k > 0).
Original entry on oeis.org
1, 3, 9, 15, 27, 37, 165, 207, 897, 5761, 16753, 17995, 96567
Offset: 1
If k = 1 then (10^1*361 - 163)/9 = 383 (prime), so 1 is a term.
If k = 9 then (10^9*361 - 163)/9 = 40111111093 (prime), so 9 is a term.
Showing 1-4 of 4 results.
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