A114655
Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.
Original entry on oeis.org
2, 4, 2, 8, 12, 2, 16, 48, 24, 2, 32, 160, 160, 40, 2, 64, 480, 800, 400, 60, 2, 128, 1344, 3360, 2800, 840, 84, 2, 256, 3584, 12544, 15680, 7840, 1568, 112, 2, 512, 9216, 43008, 75264, 56448, 18816, 2688, 144, 2, 1024, 23040, 138240, 322560, 338688, 169344
Offset: 1
T(3,3)=2 because we have (U)D(U)D(H) and (U)D(U)D(U)D, where U=(1,1), D=(1,-1) and H=(2,0) (the weak ascents are shown between parentheses).
Triangle starts:
2;
4,2;
8,12,2;
16,48,24,2;
32,160,160,40,2.
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T:=(n,k)->2^(n-k+1)*binomial(n,k)*binomial(n,k-1)/n: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
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Flatten[Table[2^(n-k+1) Binomial[n,k] Binomial[n,k-1]/n,{n,10}, {k,n}]] (* Harvey P. Dale, Oct 01 2011 *)
A087306
a(n) is the start of the first arithmetic progression with common difference n of n numbers with the same prime signature.
Original entry on oeis.org
1, 3, 155, 111, 3875983, 1333, 763274553951, 11978
Offset: 1
a(4) = 111 as 111, 115, 119 and 123 all are of the prime signature p*q.
A114687
Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).
Original entry on oeis.org
1, 1, 2, 1, 6, 4, 1, 12, 24, 8, 1, 20, 80, 80, 16, 1, 30, 200, 400, 240, 32, 1, 42, 420, 1400, 1680, 672, 64, 1, 56, 784, 3920, 7840, 6272, 1792, 128, 1, 72, 1344, 9408, 28224, 37632, 21504, 4608, 256, 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520
Offset: 1
T(3,2)=4 because we have UbUbUDDD, UbUrUDDD, UrUbUDDD and UrUrUDDD, where U=(1,1), D=(1,-1) and b (r) indicates a blue (red) double rise.
Triangle begins:
1;
1, 2;
1, 6, 4;
1, 12, 24, 8;
1, 20, 80, 80, 16.
Triangle [1,0,1,0,1,0,1,0,...] DELTA [0,2,0,2,0,2,0,2,0,...]:= T(n,k), 0 <= k <= n, begins: 1; 1,0; 1,2,0; 1,6,4,0; 1,12,24,8,0; 1,20,80,80,16,0; ... - _Philippe Deléham_, Jan 02 2009
- Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
- D. Callan, Polygon Dissections and Marked Dyck Paths
- G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 23-24.
- G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
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T:=(n,k)->2^k*binomial(n,k)*binomial(n,k+1)/n: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od;
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Table[2^k*Binomial[n, k] Binomial[n, k + 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Nov 05 2017 *)
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t(r, m) = 2^m*binomial(r, m)*binomial(r, m+1)/r;
tabl(nn) = {for (n=1, nn, for (k=0, n-1, print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Nov 22 2014
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