A026772 -id:A026772 - cp's OEIS frontend

A026769 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; T(2,1)=2; for n >= 3 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if 1<=k<=(n-1)/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 7, 4, 1, 1, 8, 17, 11, 5, 1, 1, 10, 31, 28, 16, 6, 1, 1, 12, 49, 76, 44, 22, 7, 1, 1, 14, 71, 156, 120, 66, 29, 8, 1, 1, 16, 97, 276, 352, 186, 95, 37, 9, 1, 1, 18, 127, 444, 784, 538, 281, 132, 46, 10, 1, 1, 20, 161, 668, 1504, 1674, 819, 413, 178, 56, 11, 1

Offset: 0

Clark Kimberling

T(n, k) is the number of paths from (0, 0) to (k,n-k) in the directed graph having vertices (i, j) (i and j in range [0,n]) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j>=0 and edges (i,i+h)-to-(i+1,i+h+1) for i>=0, h>=1.

Also, square array R read by antidiagonals where R(i,j) = T(i+j,i), which is equal to the number of paths from (0,0) to (i,j) in the above graph. - Max Alekseyev, Dec 02 2015

			Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   7,   4,   1;
  1,  8,  17,  11,   5,   1;
  1, 10,  31,  28,  16,   6,   1;
  1, 12,  49,  76,  44,  22,   7,   1;
  1, 14,  71, 156, 120,  66,  29,   8,  1;
  1, 16,  97, 276, 352, 186,  95,  37,  9,  1;
  1, 18, 127, 444, 784, 538, 281, 132, 46, 10, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. A. Alekseyev. On Enumeration of Dyck-Schroeder Paths. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68; arXiv:1601.06158 [math.CO], 2016-2018.

Cf. A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026779

Cf. A026780 (a variant with h>=0)

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n=2 and k=1) then return 2;
    elif (k <= Int((n-1)/2)) then return T(n-1,k-1)+T(n-2,k-1) +T(n-1,k);
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Oct 31 2019

A026769 := proc(n,k)
    option remember;
    if k= 0 or k =n then
        1;
    elif n= 2 and k= 1 then
        2;
    elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    fi ;
end proc: # R. J. Mathar, Jun 15 2014

T[n_, k_] := T[n, k] = Which[k==0 || k==n, 1, n==2 && k==1, 2, k <= (n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], True, T[n-1, k-1] + T[n-1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2017, from Maple *)

T(n,k) = if(k==0 || k==n, 1, if(n==2 && k==1, 2, if( k<=(n-1)/2, T(n-1,k-1) + T(n-2,k-1) + T(n-1,k), T(n-1,k-1) + T(n-1,k) )));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 31 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 31 2019

For n>=2*k, T(n,k) = coefficient of x^k in G(x)*S(x)^(n-2*k). For n<=2*k, T(n,k) = coefficient of x^(n-k) in G(x)*C(x)^(2*k-n). Here C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x - sqrt(1-6*x+x^2) )/(2*x) is o.g.f. for A006318, and G(x)=1/(1-x*(C(x)+S(x))) is o.g.f. for A026770. - Max Alekseyev, Dec 02 2015

Offset corrected by R. J. Mathar, Jun 15 2014

More terms added by G. C. Greubel, Oct 31 2019

A026770 a(n) = T(2n,n), T given by A026769.

Original entry on oeis.org

1, 2, 7, 28, 120, 538, 2493, 11854, 57558, 284392, 1426038, 7241356, 37173304, 192638992, 1006564439, 5297715628, 28061959428, 149491856978, 800425486692, 4305263668514, 23251846197766, 126044501870378, 685569373724964, 3740339567665558, 20463965229643218, 112250484320225118

Offset: 0

Clark Kimberling

nonn

Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when below the diagonal, (1,1). - Alois P. Heinz, Sep 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A006318, A026781, A104625, A109980.

Cf. A026769, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(x + Sqrt(1-4*x) + Sqrt(1-6*x+x^2)) )); // G. C. Greubel, Nov 01 2019

seq(coeff(series(2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_] := T[n, k] = Which[k==0 || k==n, 1, n==2 && k==1, 2, k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], True, T[n-1, k-1] + T[n-1, k]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 24 2019 *)

{ C = (1-sqrt(1-4*x+O(x^51)))/2/x; S = (1-x-sqrt(1-6*x+x^2 +O(x^51)))/2/x; Vec(1/(1-x*(C+S))) } /* Max Alekseyev, Dec 02 2015 */

def A026770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)) ).list()
A026770_list(30) # G. C. Greubel, Nov 01 2019

O.g.f.: 1/(1-x*(C(x)+S(x))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108 and S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318. - Max Alekseyev, Dec 02 2015

A026771 a(n) = T(2n,n-1), T given by A026769.

Original entry on oeis.org

1, 6, 31, 156, 784, 3962, 20173, 103522, 535294, 2787700, 14613710, 77072816, 408737760, 2178631156, 11666175215, 62734622764, 338660977020, 1834690352066, 9971834477972, 54361287536706, 297170702049966, 1628670524735842

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A026769, A026770, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(2*n, n-1), n=1..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[T[2*n, n-1], {n, 30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n, n-1) for n in (1..30)] # G. C. Greubel, Nov 01 2019

A026773 a(n) = T(2n-1,n-1), T given by A026769. Also T(2n+1,n+1), T given by A026780.

Original entry on oeis.org

1, 4, 17, 76, 352, 1674, 8129, 40156, 201236, 1020922, 5234660, 27089726, 141335846, 742712598, 3927908193, 20891799036, 111688381228, 599841215226, 3234957053984, 17512055200470, 95125188934942, 518340392855286, 2832580291316092, 15520177744727766

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A026769, A026770, A026771, A026772, A026774, A026775, A026776, A026777, A026778, A026779.

List([0..30], n-> Sum([0..n], k-> Binomial(n+1,k)*Binomial(n+k+1, n+1)/(k+1) )); # G. C. Greubel, Nov 01 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt(1-4*x) - Sqrt(1-6*x+x^2))/(2*x) -1/2 )); // G. C. Greubel, Nov 01 2019

seq(coeff(series((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2, x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 01 2019

Rest@CoefficientList[Series[(Sqrt[1-4*x] - Sqrt[1-6*x+x^2])/(2*x) -1/2, {x,0,30}], x] (* G. C. Greubel, Nov 01 2019 *)

my(x='x+O('x^30)); Vec((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2) \\ G. C. Greubel, Nov 01 2019

def A026773_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2).list()
a=A026773_list(30); a[1:] # G. C. Greubel, Nov 01 2019

From Vladeta Jovovic, Nov 23 2003: (Start)
a(n) = A006318(n) - A000108(n).
G.f.: (sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2. (End)
From Paul Barry, May 19 2005: (Start)
a(n) = Sum_{k=0..n} C(n+k+1, n+1)*C(n+1, k)/(k+1).
a(n) = Sum_{k=0..n+1} C(n+2, k)*C(n+k, n+1)/(n+2). (End)
D-finite with recurrence n*(n+1)*a(n) -n*(11*n-7)*a(n-1) +(37*n^2-95*n+54)*a(n-2) +(-49*n^2+269*n-354)*a(n-3) +6*(9*n^2-71*n+138)*a(n-4) -4*(2*n-9)*(n-5)*a(n-5)=0. - R. J. Mathar, Aug 05 2021

A026774 a(n) = T(2n-1,n-2), T given by A026769.

Original entry on oeis.org

1, 8, 49, 276, 1504, 8082, 43193, 230536, 1231484, 6591350, 35369380, 190329098, 1027180798, 5559635866, 30176648513, 164237973028, 896188159820, 4902187071922, 26877397858264, 147684225578318, 813159429830590

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A026769, A026770, A026771, A026772, A026773, A026775, A026776, A026777, A026778, A026779.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(2*n-1, n-2), n=2..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[T[2*n-1, n-2], {n,2,30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(2*n-1, n-2) for n in (2..30)] # G. C. Greubel, Nov 01 2019

A026775 a(n) = T(n, floor(n/2)), T given by A026769.

Original entry on oeis.org

1, 1, 2, 4, 7, 17, 28, 76, 120, 352, 538, 1674, 2493, 8129, 11854, 40156, 57558, 201236, 284392, 1020922, 1426038, 5234660, 7241356, 27089726, 37173304, 141335846, 192638992, 742712598, 1006564439, 3927908193, 5297715628

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026776, A026777, A026778, A026779.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(T(n, floor(n/2)), n=0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[T[n, Floor[n/2]], {n,0,30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[T(n, floor(n/2)) for n in (0..30)] # G. C. Greubel, Nov 01 2019

A026776 a(n) = Sum_{k=0..n} T(n,k), T given by A026769.

Original entry on oeis.org

1, 2, 4, 9, 19, 43, 93, 212, 466, 1070, 2382, 5506, 12386, 28800, 65356, 152745, 349183, 819639, 1885361, 4441719, 10270279, 24269629, 56363319, 133529869, 311255601, 738947515, 1727873793, 4109314729, 9634406661, 22946573863

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026777, A026778, A026779.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(add(T(n, k), k=0..n), n=0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[Sum[T[n,k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 01 2019

A026777 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026769.

Original entry on oeis.org

1, 1, 3, 5, 14, 26, 70, 138, 362, 742, 1912, 4028, 10249, 22033, 55547, 121273, 303641, 670997, 1671233, 3729071, 9250099, 20803231, 51437219, 116436313, 287152067, 653567143, 1608416195, 3677760541, 9035150126, 20741496354

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026778, A026779.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[Sum[T[n,k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Nov 01 2019

A026779 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026769.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 32, 56, 97, 181, 322, 567, 1053, 1892, 3369, 6241, 11286, 20255, 37463, 68044, 122809, 226896, 413376, 749159, 1382990, 2525162, 4590351, 8468738, 15487526, 28218889, 52035094, 95273724, 173898941

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(add(T(n-k,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[Sum[T[n-k,k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(T(n-k,k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Nov 01 2019

A026778 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026769.

Original entry on oeis.org

1, 3, 7, 16, 35, 78, 171, 383, 849, 1919, 4301, 9807, 22193, 50993, 116349, 269094, 618277, 1437916, 3323277, 7764996, 18035275, 42304904, 98668223, 232198092, 543453693, 1282401208, 3010275001, 7119589730, 16753996391

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026779.

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k=n then 1;
   elif n=2 and k=1 then 2;
   elif k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(add(add(T(j,k), k=0..n), j=0..n), n=0..30); # G. C. Greubel, Nov 01 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[Sum[T[j,k], {k,0,n}, {j,0,n}], {n,0,30}] (* G. C. Greubel, Nov 01 2019 *)

@CachedFunction
def T(n, k):
    if n < 0:
        return 0
    elif (k==0 or k==n): return 1
    elif (n==2 and k==1): return 2
    elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
    else: return T(n-1,k-1) + T(n-1,k)
[sum(sum(T(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 01 2019

cp's OEIS Frontend

A026769 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; T(2,1)=2; for n >= 3 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if 1<=k<=(n-1)/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

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A026770 a(n) = T(2n,n), T given by A026769.

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A026771 a(n) = T(2n,n-1), T given by A026769.

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A026773 a(n) = T(2n-1,n-1), T given by A026769. Also T(2n+1,n+1), T given by A026780.

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A026774 a(n) = T(2n-1,n-2), T given by A026769.

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A026775 a(n) = T(n, floor(n/2)), T given by A026769.

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A026776 a(n) = Sum_{k=0..n} T(n,k), T given by A026769.

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