cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Clark Kimberling

Keywords

Comments

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

Links

Crossrefs

Cf. A000108, A006318, A026781, A104625, A109980.
Cf. A026769, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

Programs

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

Formula

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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

A026772 a(n) = T(2n, n-2), T given by A026769.

Original entry on oeis.org

1, 10, 71, 444, 2616, 14938, 83821, 465654, 2572166, 14164320, 77886902, 428113940, 2353823912, 12950837432, 71326701751, 393289209772, 2171308560036, 12003376308370, 66445540183348, 368304502202306, 2044177115127750
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026769, A026770, A026771, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

Programs

  • Maple
    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-2), n=2..30); # G. C. Greubel, Nov 01 2019
  • Mathematica
    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-2], {n,2,30}] (* G. C. Greubel, Nov 01 2019 *)
  • Sage
    @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-2) for n in (2..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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

Formula

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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
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