A127538
Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k odd-length branches starting at the root (0<=k<=n).
Original entry on oeis.org
1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 3, 3, 7, 0, 1, 3, 22, 6, 10, 0, 1, 16, 43, 50, 9, 13, 0, 1, 37, 175, 101, 87, 12, 16, 0, 1, 134, 503, 448, 177, 133, 15, 19, 0, 1, 411, 1784, 1305, 862, 271, 188, 18, 22, 0, 1, 1411, 5887, 4848, 2524, 1444, 383, 252, 21, 25, 0, 1, 4747, 20604
Offset: 0
T(2,2)=1 because we have the tree /\.
Triangle starts:
1;
0,1;
1,0,1;
0,4,0,1;
3,3,7,0,1;
3,22,6,10,0,1;
-
C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z^2*C-t*z*C): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
A127541
Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n).
Original entry on oeis.org
1, 1, 1, 1, 3, 2, 8, 5, 1, 24, 15, 3, 75, 46, 10, 1, 243, 148, 34, 4, 808, 489, 116, 16, 1, 2742, 1652, 402, 61, 5, 9458, 5678, 1408, 228, 23, 1, 33062, 19792, 4982, 847, 97, 6, 116868, 69798, 17783, 3138, 393, 31, 1, 417022, 248577, 63967, 11627, 1557, 143, 7
Offset: 0
T(2,0)=1 because we have the tree /\.
Triangle starts:
1;
1;
1,1;
3,2;
8,5,1;
24,15,3;
-
C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z*C-t*z^2*C): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
A168377
Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.
Original entry on oeis.org
1, -1, 1, 1, 0, 1, -1, 2, 1, 1, 1, 3, 4, 2, 1, -1, 11, 10, 7, 3, 1, 1, 31, 32, 21, 11, 4, 1, -1, 101, 100, 69, 37, 16, 5, 1, 1, 328, 329, 228, 128, 59, 22, 6, 1, -1, 1102, 1101, 773, 444, 216, 88, 29, 7, 1, 1, 3760, 3761, 2659, 1558, 785, 341, 125, 37, 8, 1
Offset: 0
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
-1, 1;
1, 0, 1;
-1, 2, 1, 1;
1, 3, 4, 2, 1;
-1, 11, 10, 7, 3, 1;
1, 31, 32, 21, 11, 4, 1;
-1, 101, 100, 69, 37, 16, 5, 1;
...
From _Philippe Deléham_, Sep 14 2014: (Start)
Production matrix begins:
-1, 1
0, 1, 1
0, 1, 1, 1
0, 1, 1, 1, 1
0, 1, 1, 1, 1, 1
0, 1, 1, 1, 1, 1, 1
0, 1, 1, 1, 1, 1, 1, 1
0, 1, 1, 1, 1, 1, 1, 1, 1
... (End)
- Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
- Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, Annals of Combinatorics, 13 (2009), 65-85.
- L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Applied Mathematics, 34(1-3) (1991), 229-239.
- Wikipedia, Riordan array.
Cf.
A000012,
A000108,
A000124,
A023443,
A032357,
A033297,
A033999,
A091491,
A096470,
A106566,
A127540.
-
A000108(n) = binomial(2*n, n)/(n+1);
A032357(n) = sum(k=0, n, (-1)^(n-k)*A000108(k));
T(n, k) = if ((k==0), (-1)^n, if ((n<0) || (k<0), 0, if (k==1, A032357(n-1), if (n > k-1, T(n, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020
Showing 1-3 of 3 results.
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