A014151 -id:A014151 - cp's OEIS frontend

A185944 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m = 3.

1, 3, 1, 6, 4, 1, 10, 11, 5, 1, 15, 27, 17, 6, 1, 21, 66, 51, 24, 7, 1, 28, 170, 149, 83, 32, 8, 1, 36, 471, 443, 273, 124, 41, 9, 1, 45, 1398, 1362, 891, 448, 175, 51, 10, 1, 55, 4381, 4336, 2938, 1576, 685, 237, 62, 11, 1, 66, 14282, 14227, 9846, 5510, 2572, 996, 311, 74, 12, 1

Offset: 0

Vladimir Kruchinin, Feb 07 2011

			Array begins
   1;
   3,   1;
   6,   4,   1;
  10,  11,   5,   1;
  15,  27,  17,   6,   1;
  21,  66,  51,  24,   7,   1;
  28, 170, 149,  83,  32,   8,  1;
  36, 471, 443, 273, 124,  41,  9,   1;
Production matrix begins:
   3, 1;
  -3, 1, 1;
   4, 1, 1, 1;
  -2, 1, 1, 1, 1;
   1, 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;
   ... _Philippe Deléham_, Sep 20 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A091491 (m=1), A185943 (m=2), A185945 (m=4), A014151 (column k=1).

Cf. A000108, A000217.

r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 3] = (n + 1)*(n + 2)/2; Table[ r[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

R(n,k,m) = k*Sum_(i=0..n-k,binomial(i+m-1,m-1)*binomial(2*(n-i)-k-1,n-i-1)/(n-i)), m=3, k>0.

R(n,0,3) = (n+1)*(n+2)/2 = A000217(n+1).

A101974 Triangle read by rows: number of Dyck paths of semilength n with k peaks before the first return (1<= k

Original entry on oeis.org

1, 2, 4, 1, 9, 4, 1, 23, 11, 7, 1, 65, 27, 28, 11, 1, 197, 66, 87, 62, 16, 1, 626, 170, 239, 250, 122, 22, 1, 2056, 471, 627, 829, 630, 219, 29, 1, 6918, 1398, 1656, 2448, 2553, 1419, 366, 37, 1, 23714, 4381, 4554, 6803, 8813, 6979, 2917, 578, 46, 1, 82500, 14282

Offset: 1

Emeric Deutsch, Dec 22 2004

			T(4,2)=4 because we have U(UD)(UD)D|UD, U(UD)U(UD)DD|, UU(UD)D(UD)D| and
UU(UD)(UD)DD|, where U=(1,1), D=(1,-1) (the peaks before the first return | are shown between parentheses).
     1
       2
     4      1
     9      4      1
    23     11      7      1
    65     27     28     11      1
   197     66     87     62     16      1
   626    170    239    250    122     22      1
  2056    471    627    829    630    219     29      1
  6918   1398   1656   2448   2553   1419    366     37      1
 23714   4381   4554   6803   8813   6979   2917    578     46      1
 82500  14282  13231  18571  27362  28364  17206   5567    872     56      1

E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.

Cf. A000108 (row sums), A014137 (column k=1), A014151 (column k=2), A101975.

c:=n->binomial(2*n,n)/(n+1):
T:=proc(n,k) if k=1 then sum(c(i),i=0..n-1) else sum(c(j)*binomial(n-1-j,k-1)*binomial(n-1-j,k)/(n-1-j),j=0..n-2) fi end proc:
T(1,1);
for n from 1 to 12 do seq(T(n,k),k=1..n-1) od; # yields the sequence in triangular form

T(n, 1)=sum(c(i), i=0..n-1), T(n, k)=sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) for k>1, where c(i)=binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108);

G.f.=1+tzC(z)[1+r(t, z)], where C(z)=1+zC(z)^2 is the Catalan function and r(t, z)=z[1+r(t, z)][1+tr(t, z)] is the Narayana function.

cp's OEIS Frontend

A185944 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m = 3.

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