A026754 -id:A026754 - cp's OEIS frontend

A026747 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1 <= k <= n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is even and 1 <= k <= n/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 5, 1, 1, 7, 17, 16, 6, 1, 1, 9, 30, 44, 22, 7, 1, 1, 10, 39, 74, 66, 29, 8, 1, 1, 12, 58, 143, 184, 95, 37, 9, 1, 1, 13, 70, 201, 327, 279, 132, 46, 10, 1, 1, 15, 95, 329, 671, 790, 411, 178, 56, 11, 1, 1, 16, 110, 424, 1000, 1461, 1201, 589, 234, 67, 12, 1

Offset: 0

Clark Kimberling

			Triangle begins as:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  4,  1;
  1, 6, 11,  5,  1;
  1, 7, 17, 16,  6, 1;
  1, 9, 30, 44, 22, 7, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A026754 (row sums).

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n mod 2)=0 and k List([0..n], k-> T(n,k) ))); # G. C. Greubel, Oct 28 2019

A026747 := proc(n,k)
    if k=0 or k = n then
        1;
    elif type(n,'even') and k <= n/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: # R. J. Mathar, Jun 30 2013

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && 1<=k<=n/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, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 28 2019 *)

T(n,k) = if(k==0 || k==n, 1, if(n%2==0 && k<=n/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 28 2019

def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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 28 2019

T(n, k) = number of paths from (0, 0) to (n-k, k) in the directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, 2h+i)-to-(i+1, 2h+i+1) for i >= 0, h>=0.

More terms added by G. C. Greubel, Oct 28 2019

A026749 a(n) = T(2n,n-1), T given by A026747.

Original entry on oeis.org

1, 6, 30, 143, 671, 3132, 14601, 68101, 318035, 1487661, 6971222, 32727472, 153926409, 725264305, 3423262180, 16185240446, 76648901377, 363557014067, 1726994886004, 8215502584008, 39135887682617, 186676023857041, 891557875400175

Offset: 1

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026750, A026751, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(2*n,n-1), n=1..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[2n, n-1], {n,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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, Oct 29 2019

A026748 a(n) = T(2n,n), T given by A026747.

Original entry on oeis.org

1, 3, 11, 44, 184, 790, 3452, 15278, 68290, 307696, 1395696, 6367199, 29193025, 134442102, 621609060, 2884432810, 13428450520, 62703991531, 293606387095, 1378309455352, 6485734373020, 30586630485443, 144544075759391, 684395988590939

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A026747, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(2*n,n), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[2n, n], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026750 a(n) = T(2n,n-2), T given by A026747.

Original entry on oeis.org

1, 9, 58, 329, 1753, 9020, 45442, 225860, 1112543, 5446607, 26550968, 129042976, 625860205, 3031021096, 14664729519, 70906318405, 342717456708, 1656208470644, 8003645557573, 38681730323747, 186985728069661, 904119336235884

Offset: 2

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026751, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(2*n,n-2), n=2..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[2n, n-2], {n,2,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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-2) for n in (2..30)] # G. C. Greubel, Oct 29 2019

A026751 a(n) = T(2n-1,n-1), T given by A026747.

Original entry on oeis.org

1, 4, 17, 74, 327, 1461, 6584, 29879, 136391, 625731, 2883357, 13338421, 61920497, 288368511, 1346873365, 6307694990, 29613690966, 139352892908, 657163401162, 3105304341356, 14701236957028, 69722518168060, 331220099616432

Offset: 1

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026750, A026752, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(2*n-1,n-1), n=1..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[2n-1, n-1], {n,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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-1) for n in (1..30)] # G. C. Greubel, Oct 29 2019

A026752 a(n) = T(2n-1,n-2), T given by A026747.

Original entry on oeis.org

1, 7, 39, 201, 1000, 4885, 23621, 113543, 543895, 2600204, 12417829, 59278440, 282969385, 1351124510, 6454283276, 30849969965, 147555219782, 706274470775, 3383203356648, 16219148141581, 77817618006364, 373661751926702

Offset: 2

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026750, A026751, A026753, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(2*n-1,n-2), n=2..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[2n-1, n-2], {n,2,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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, Oct 29 2019

A026753 a(n) = T(n, floor(n/2)), T given by A026747.

Original entry on oeis.org

1, 1, 3, 4, 11, 17, 44, 74, 184, 327, 790, 1461, 3452, 6584, 15278, 29879, 68290, 136391, 307696, 625731, 1395696, 2883357, 6367199, 13338421, 29193025, 61920497, 134442102, 288368511, 621609060, 1346873365, 2884432810

Offset: 0

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026754, A026755, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(n,floor(n/2)), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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, Oct 29 2019

A026755 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026747.

Original entry on oeis.org

1, 1, 4, 5, 18, 25, 84, 124, 398, 612, 1901, 3012, 9126, 14800, 43968, 72658, 212417, 356544, 1028520, 1749344, 4989477, 8583258, 24244139, 42121079, 117973702, 206754379, 574811040, 1015179978, 2803969443, 4986329826

Offset: 0

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026756, A026757.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(n,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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], Floor[n/2]], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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, Oct 29 2019

A026756 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026747.

Original entry on oeis.org

1, 3, 8, 18, 42, 90, 204, 432, 972, 2052, 4610, 9726, 21859, 46125, 103783, 219099, 493699, 1042899, 2353716, 4975350, 11247138, 23790714, 53867343, 114020601, 258571256, 547672566, 1243848888, 2636201532, 5995717860

Offset: 0

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026757.

A026747 := proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(i,j), j=0..n), i=0..n), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[i, j], {i,0,n},{j,0,n}], {n,0,30}] (* G. C. Greubel, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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(i,j) for j in (0..n)) for i in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 29 2019

A026757 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026747.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 20, 32, 58, 102, 169, 302, 527, 888, 1573, 2741, 4661, 8215, 14316, 24481, 43023, 74998, 128747, 225867, 393838, 678047, 1188201, 2072239, 3575728, 6261248, 10921278, 18879372, 33040083, 57637061

Offset: 0

Clark Kimberling

nonn

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

Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026756.

A026747 := proc(n,k) option remember;
   if k=0 or k = n then 1;
   elif type(n,'even') and k <= n/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(A026747(n-k,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 29 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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, Oct 29 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k<=n/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, Oct 29 2019

cp's OEIS Frontend

A026747 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1 <= k <= n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is even and 1 <= k <= n/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Sage

Formula

Extensions

A026749 a(n) = T(2n,n-1), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026748 a(n) = T(2n,n), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026750 a(n) = T(2n,n-2), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026751 a(n) = T(2n-1,n-1), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026752 a(n) = T(2n-1,n-2), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026753 a(n) = T(n, floor(n/2)), T given by A026747.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A026755 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026747.

Views

Author

Keywords