A166830 -id:A166830 - cp's OEIS frontend

A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 2, 5, 4, 1, 0, 0, 0, 2, 8, 9, 5, 1, 0, 0, 0, 1, 12, 18, 14, 6, 1, 0, 0, 0, 1, 17, 34, 33, 20, 7, 1, 0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1, 0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1, 0, 0, 0, 0, 41, 187, 301, 303, 221, 118, 44, 10, 1

Offset: 1

Alois P. Heinz, Jul 31 2013

			:   T(6,4) = 3              :  T(11,3) = 1  :
:     o       o       o     :        o      :
:    / \      |       |     :      /( )\    :
:   o   o     o       o     :     o o o o   :
:   |        / \      |     :    /| | |     :
:   o       o   o     o     :   o o o o     :
:   |       |        / \    :   |   |       :
:   o       o       o   o   :   o   o       :
:   |       |       |       :               :
:   o       o       o       :               :
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 0, 2,  1;
  0, 0, 0, 2,  3,   1;
  0, 0, 0, 2,  5,   4,   1;
  0, 0, 0, 2,  8,   9,   5,   1;
  0, 0, 0, 1, 12,  18,  14,   6,  1;
  0, 0, 0, 1, 17,  34,  33,  20,  7,  1;
  0, 0, 0, 1, 23,  61,  72,  54, 27,  8, 1;
  0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.
Row sums give: A004111.
Column sums give: A038081.
Largest n with T(n,k)>0 is A038093(k).
Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.
T(2n,n) gives A245090.
T(2n+1,n) gives A245091.
Cf. A034781.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
      add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..15);

Drop[Transpose[Map[PadRight[#,15]&,Table[f[n_]:=Nest[ CoefficientList[ Series[ Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,15}],x]&,{1},n]; f[m]-PadRight[f[m-1],Length[f[m]]],{m,1,15}]]],1]//Grid (* Geoffrey Critzer, Aug 01 2013 *)

A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 6, 3, 1, 24, 18, 10, 4, 1, 75, 57, 33, 15, 5, 1, 243, 186, 111, 54, 21, 6, 1, 808, 622, 379, 193, 82, 28, 7, 1, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1, 33062, 25724, 16266, 8928, 4332, 1856, 692, 218, 55, 10, 1

Offset: 2

N. J. A. Sloane, Dec 02 2001

A Riordan triangle.
Subtriangle of triangle in A167772. - Philippe Deléham, Nov 14 2009
Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A000958 and g(x) is the g.f. of A000108. - Philippe Deléham, Jan 23 2010

			T(3,2)=1 reflecting the unique Dyck path (UUDUDD) of length 6, with no hills and height of first peak equal to 2.
Triangle begins:
     1;
     1,    1;
     3,    2,    1;
     8,    6,    3,   1;
    24,   18,   10,   4,   1;
    75,   57,   33,  15,   5,   1;
   243,  186,  111,  54,  21,   6,  1;
   808,  622,  379, 193,  82,  28,  7,  1;
  2742, 2120, 1312, 690, 311, 118, 36,  8,  1;

Reinhard Zumkeller, Rows n = 2..125 of triangle, flattened
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Row sums give A000957 (the Fine sequence).
First column is A000958.
Cf. A000108, A007318, A167772.

a065602 n k = sum
   [(k-1+2*j) * a007318' (2*n-k-1-2*j) (n-1) `div` (2*n-k-1-2*j) |
    j <- [0 .. div (n-k) 2]]
a065602_row n = map (a065602 n) [2..n]
a065602_tabl = map a065602_row [2..]
-- Reinhard Zumkeller, May 15 2014

a := proc(n,k) if n=0 and k=0 then 1 elif k<2 or k>n then 0 else sum((k-1+2*j)*binomial(2*n-k-1-2*j,n-1)/(2*n-k-1-2*j),j=0..floor((n-k)/2)) fi end: seq(seq(a(n,k),k=2..n),n=1..14);

nmax = 12; t[n_, k_] := Sum[(k-1+2j)*Binomial[2n-k-1-2j, n-1] / (2n-k-1-2j), {j, 0, (n-k)/2}]; Flatten[ Table[t[n, k], {n, 2, nmax}, {k, 2, n}]] (* Jean-François Alcover, Nov 08 2011, after Maple *)

def T(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
flatten([[T(n,k) for k in (2..n)] for n in (2..12)]) # G. C. Greubel, May 26 2022

T(n, 2) = A000958(n-1).
Sum_{k=2..n} T(n, k) = A000957(n+1).
From Emeric Deutsch, Feb 23 2004: (Start)
T(n, k) = Sum_{j=0..floor((n-k)/2)} (k-1+2*j)*binomial(2*n-k-1-2*j, n-1)/(2*n-k-1-2*j).
G.f.: t^2*z^2*C/( (1-z^2*C^2)*(1-t*z*C) ), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
T(n,k) = A167772(n-1,k-1), k=2..n. - Reinhard Zumkeller, May 15 2014
From G. C. Greubel, May 26 2022: (Start)
T(n, n-1) = A000027(n-2).
T(n, n-2) = A000217(n-2).
T(n, n-3) = A166830(n-3). (End)

More terms from Emeric Deutsch, Feb 23 2004

A167772 Riordan array (c(x)/(1+xc(x)), xc(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 6, 8, 6, 3, 1, 18, 24, 18, 10, 4, 1, 57, 75, 57, 33, 15, 5, 1, 186, 243, 186, 111, 54, 21, 6, 1, 622, 808, 622, 379, 193, 82, 28, 7, 1, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 7338, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1

Offset: 0

Philippe Deléham, Nov 11 2009, corrected Nov 12 2009

			Triangle begins:
     1;
     0,    1;
     1,    1,    1;
     2,    3,    2,    1;
     6,    8,    6,    3,   1;
    18,   24,   18,   10,   4,   1;
    57,   75,   57,   33,  15,   5,   1;
   186,  243,  186,  111,  54,  21,   6,  1;
   622,  808,  622,  379, 193,  82,  28,  7,  1;
  2120, 2742, 2120, 1312, 690, 311, 118, 36,  8,  1;
Production matrix begins:
  0, 1;
  1, 1, 1;
  1, 1, 1, 1;
  1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ... - _Philippe Deléham_, Mar 03 2013

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

Cf. A000957, A000958 (row sums), A001558, A001559, A104629, A167769.

Diagonals: A000012, A000217, A001477, A166830.

import Data.List (genericIndex)
a167772 n k = genericIndex (a167772_row n) k
a167772_row n = genericIndex a167772_tabl n
a167772_tabl = [1] : [0, 1] :
               map (\xs@(:x:) -> x : xs) (tail a065602_tabl)
-- Reinhard Zumkeller, May 15 2014

A065602[n_, k_]:= A065602[n,k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n-j) -k-1), {j, 0, (n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n+1,3] + Boole[n==0], A065602[n+1, k+1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2022 *)

def A065602(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A167772(n,k):
    if (k==0): return A065602(n+1,3) + bool(n==0)
    else: return A065602(n+1,k+1)
flatten([[A167772(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2022

Sum_{k=0..n} T(n, k) = A000958(n+1).
From Philippe Deléham, Nov 12 2009: (Start)
Sum_{k=0..n} T(n,k)*2^k = A014300(n).
Sum_{k=0..n} T(n,k)*2^(n-k) = A064306(n). (End)
For n > 0: T(n,0) = A065602(n+1,3), T(n,k) = A065602(n+1,k+1), k = 1..n. - Reinhard Zumkeller, May 15 2014

A209704 Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.

Original entry on oeis.org

1, 3, 1, 4, 3, 2, 5, 6, 8, 3, 6, 10, 18, 14, 5, 7, 15, 33, 38, 27, 8, 8, 21, 54, 81, 83, 49, 13, 9, 28, 82, 150, 197, 170, 89, 21, 10, 36, 118, 253, 401, 448, 342, 159, 34, 11, 45, 163, 399, 736, 999, 987, 671, 282, 55, 12, 55, 218, 598, 1253, 1988, 2387, 2106

Offset: 1

Clark Kimberling, Mar 12 2012

For n>1, row n starts with n+1, followed by the n-th
triangular number, and ends in F(n+1), where F=A000045
(Fibonacci numbers).
Column 3:  A166830.
Row sums:  A048654.
Alternating row sums: 1,2,3,4,5,6,7,8,9,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3...1
4...3....2
5...6....8....3
6...10...18...14...5
First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 2x^2.

Cf. A209703, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A209703 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A209704 *)

u(n,x)=x*u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 1, 1, 1, -2, 2, 3, 2, 1, -6, 6, 8, 6, 3, 1, -18, 18, 24, 18, 10, 4, 1, -57, 57, 75, 57, 33, 15, 5, 1, -186, 186, 243, 186, 111, 54, 21, 6, 1, -622, 622, 808, 622, 379, 193, 82, 28, 7, 1, -2120, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1

Offset: 0

Philippe Deléham, Feb 10 2014

			Triangle begins:
    1;
   -1,  1;
    0,  0,  1;
   -1,  1,  1,  1;
   -2,  2,  3,  2,  1;
   -6,  6,  8,  6,  3,  1;
  -18, 18, 24, 18, 10,  4, 1;
  -57, 57, 75, 57, 33, 15, 5, 1;
Production matrix begins:
  -1, 1;
  -1, 1, 1;
  -1, 1, 1, 1;
  -1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Diagonals: A000012, A000217, A023443, A166830.
Cf. A000957, A000958, A001558, A001559, A104629, A126983.
Cf. A065602, A167772.

A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j,0,(n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 27 2022 *)

def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A237619(n, k):
    if (n<2): return (-1)^(n-k)
    elif (k==0): return A065602(n, 0)
    else: return A065602(n, k)
flatten([[A237619(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2022

Sum_{k=0..n} T(n,k)*x^k = A126983(n), A000957(n+1), A026641(n) for x = 0, 1, 2 respectively.
T(n, k) = A167772(n-1, k-1) for k > 0, with T(n, 0) = A167772(n, 0).
T(n, 0) = A126983(n).
T(n+1, 1) = A000957(n+1).
T(n+2, 2) = A000958(n+1).
T(n+3, 3) = A104629(n) = A000957(n+3).
T(n+4, 4) = A001558(n).
T(n+5, 5) = A001559(n).
T(n, k) = A065602(n, k) for k > 0, with T(n, k) = (-1)^(n-k), for n < 2, and T(n, 0) = A065602(n, 0). - G. C. Greubel, May 27 2022

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A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

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A167772 Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.

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A209704 Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.

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A237619 Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).

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A167772 Riordan array (c(x)/(1+xc(x)), xc(x)), c(x) the g.f. of A000108.

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).