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 1-10 of 10 results.

A026780 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 1 <= k <= floor(n/2), else T(n,k) = T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 12, 5, 1, 1, 9, 24, 17, 6, 1, 1, 11, 40, 53, 23, 7, 1, 1, 13, 60, 117, 76, 30, 8, 1, 1, 15, 84, 217, 246, 106, 38, 9, 1, 1, 17, 112, 361, 580, 352, 144, 47, 10, 1, 1, 19, 144, 557, 1158, 1178, 496, 191, 57, 11, 1
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

T(n,k) is the number of paths from (0,0) to (k,n-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,i+h)-to-(i+1,i+h+1) for i>=0, h>=0.
Also, square array R read by antidiagonals with R(i,j) = T(i+j,i) equal number of paths from (0,0) to (i,j). - Max Alekseyev, Jan 13 2015

Examples

			The array T(n,k) starts with:
n=0: 1;
n=1: 1,  1;
n=2: 1,  3,  1;
n=3: 1,  5,  4,  1;
n=4: 1,  7, 12,  5,  1;
n=5: 1,  9, 24, 17,  6, 1;
n=6: 1, 11, 40, 53, 23, 7, 1;
...

Links

Crossrefs

Cf. A026787 (row sums), A026781 (center elements), A249488 (row-reversed version).
Cf. A026782, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

Programs

  • GAP
    T:= function(n,k)
    if n<0 then return 0;
    elif k=0 or k=n then return 1;
    elif (k <= Int(n/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
  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq(seq(T(n,k), k=0..n), n=0..12); # G. C. Greubel, Nov 01 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 01 2019 *)
  • PARI
    T(n,k) = if(n<0, 0, if(k==0 || k==n, 1, if( 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 31 2019
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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 31 2019

Formula

For n>=2*k, T(n,k) = coefficient of x^k in F(x)*S(x)^(n-2*k). For n<=2*k, T(n,k) = coefficient of x^(n-k) in F(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 F(x) = S(x)/(1 - x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015

Extensions

Edited by Max Alekseyev, Dec 02 2015

A026781 a(n) = T(2n,n), T given by A026780.

Original entry on oeis.org

1, 3, 12, 53, 246, 1178, 5768, 28731, 145108, 741392, 3825418, 19907156, 104370554, 550816506, 2924018194, 15603778253, 83661779470, 450479003038, 2435009205992, 13208558795146, 71879906857596, 392320357251928, 2147102400154768, 11780181236675858, 64782405317073968, 357022158144941548
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Number of paths from (0,0) to (n,n) 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,i+h)-to-(i+1,i+h+1) for i>=0, h>=0.

Links

Crossrefs

Cf. A026671.
Cf. A026780, A026782, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2*(1-x -Sqrt(1-6*x+x^2))/(4*x -(1 -Sqrt(1-4*x))*(1 -x -Sqrt(1-6*x+x^2))) )); // G. C. Greubel, Nov 02 2019
  • Maple
    seq(coeff(series(2*(1-x -sqrt(1-6*x+x^2))/(4*x -(1 -sqrt(1-4*x))*(1 -x -sqrt(1-6*x+x^2))), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    CoefficientList[Series[2*(1-x -Sqrt[1-6*x+x^2])/(4*x -(1 -Sqrt[1-4*x])*(1 -x -Sqrt[1-6*x+x^2])), {x,0,30}], x] (* G. C. Greubel, Nov 02 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(S/(1-x*C*S)) /* Max Alekseyev, Jan 13 2015 */
  • Sage
    def A026781_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(2*(1-x -sqrt(1-6*x+x^2))/(4*x -(1 -sqrt(1-4*x))*(1 -x -sqrt(1-6*x+x^2)))).list()
A026781_list(30) # G. C. Greubel, Nov 02 2019

Formula

O.g.f.: S(x)/(1-x*C(x)*S(x)) = (S(x)-C(x))/(x*C(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, Jan 13 2015
D-finite with recurrence 2*n*(132*n-445)*(n+2)*(n+1)*a(n) -n*(n+1) *(5587*n^2 -23082*n +12800)*a(n-1) +2*n*(n-1)*(22870*n^2 -114505*n +116854)*a(n-2) +2*(-90081*n^4 +818062*n^3 -2626791*n^2 +3517598*n -1622544)*a(n-3) +4*(85519*n^4 -1071535*n^3 +4986308*n^2 -10177616*n +7647024)*a(n-4) +(-269235*n^4 +4490125*n^3 -27985152*n^2 +77217236*n -79534224)*a(n-5) +4*(2*n-11)*(8203*n^3 -117312*n^2 +557264*n -879984)*a(n-6) -4*(n-6)*(307*n -1414) *(2*n-11) *(2*n-13)*a(n-7)=0. - R. J. Mathar, Feb 20 2020

Extensions

More terms from Max Alekseyev, Jan 13 2015

A026787 a(n) = Sum_{k=0..n} T(n,k), T given by A026780.

Original entry on oeis.org

1, 2, 5, 11, 26, 58, 136, 306, 717, 1625, 3813, 8697, 20451, 46909, 110563, 254855, 602042, 1393746, 3299304, 7666786, 18182976, 42391546, 100704606, 235452416, 560147414, 1312916040, 3127406812, 7346213746, 17518138314, 41228281888, 98408997716, 231990850378, 554207752781, 1308436686305, 3128033585157
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq( add(T(n,k), k=0..n), n=0..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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,0,n}], {n,0,30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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..n)) for n in (0..30)] # G. C. Greubel, Nov 02 2019

Formula

O.g.f.: F(x^2)*(1/(1-x*S(x^2))+C(x^2)*x/(1-x*C(x^2))), where 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 F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015
C(x^2)/(1-x*C(x^2)) above is the o.g.f. for A001405. 1/(1-x*S(x^2)) above is the o.g.f for A026003 starting with an additional 1: 1,1,1,3,5,13,25,... - R. J. Mathar, Feb 10 2022

Extensions

More terms from Max Alekseyev, Jan 13 2015

A026782 a(n) = T(2n,n-1), T given by A026780.

Original entry on oeis.org

1, 7, 40, 217, 1158, 6150, 32656, 173719, 926664, 4958556, 26619438, 143365880, 774562478, 4197344582, 22810572062, 124300860689, 679081142350, 3718894341450, 20412141531664, 112276061739814, 618806031336236, 3416954495002676
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq(T(2*n,n-1), n=1..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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[2*n, n-1], {n, 30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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, Nov 02 2019

A026783 a(n) = T(2n, n-2), T given by A026780.

Original entry on oeis.org

1, 11, 84, 557, 3446, 20514, 119336, 684227, 3886460, 21939528, 123347842, 691644044, 3871738018, 21652138770, 121026492186, 676391629701, 3780636102222, 21137831159462, 118234019051048, 661686074145618, 3705252204960252
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026782, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq(T(2*n,n-2), n=2..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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[2*n, n-2], {n,2,30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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, Nov 02 2019

A026784 a(n) = T(2n-1, n-1), T given by A026780.

Original entry on oeis.org

1, 5, 24, 117, 580, 2916, 14834, 76221, 395048, 2063104, 10847078, 57373672, 305110106, 1630489090, 8751851866, 47166202181, 255128842340, 1384688987728, 7538592535170, 41159292861980, 225315261459390, 1236441650047554
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026782, A026783, A026785, A026786, A026787, A026788, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq(T(2*n-1,n-1), n=1..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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[2*n-1, n-1], {n, 30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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, Nov 02 2019

A026785 a(n) = T(2n-1, n-2), T given by A026780.

Original entry on oeis.org

1, 9, 60, 361, 2076, 11672, 64842, 357897, 1968788, 10813804, 59372770, 326086492, 1792293014, 9861375614, 54324086446, 299651439321, 1655124211372, 9154654655044, 50704627346170, 281214708137032, 1561706813618886
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026782, A026783, A026784, A026786, A026787, A026788, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq(T(2*n-1,n-2), n=2..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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[2*n-1, n-2], {n,2,30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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, Nov 02 2019

A026786 a(n) = T(n, floor(n/2)), T given by A026780.

Original entry on oeis.org

1, 1, 3, 5, 12, 24, 53, 117, 246, 580, 1178, 2916, 5768, 14834, 28731, 76221, 145108, 395048, 741392, 2063104, 3825418, 10847078, 19907156, 57373672, 104370554, 305110106, 550816506, 1630489090, 2924018194, 8751851866
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026787, A026788, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq(T(n, floor(n/2)), n=0..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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, Nov 02 2019

A026788 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026780.

Original entry on oeis.org

1, 1, 4, 6, 20, 34, 105, 191, 563, 1071, 3057, 6007, 16745, 33729, 92332, 189662, 511812, 1068178, 2849404, 6025594, 15921514, 34043204, 89242582, 192621212, 501574732, 1091400122, 2825710822, 6192005260, 15952433940, 35172854946
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026787, A026789, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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, 0, Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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, Nov 02 2019

A026789 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026780.

Original entry on oeis.org

1, 3, 8, 19, 45, 103, 239, 545, 1262, 2887, 6700, 15397, 35848, 82757, 193320, 448175, 1050217, 2443963, 5743267, 13410053, 31593029, 73984575, 174689181, 410141597, 970289011, 2283205051, 5410611863, 12756825609, 30274963923
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Partial sums of A026787.
Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026790.

Programs

  • Maple
    T:= proc(n,k) option remember;
    if n<0 then 0;
    elif k=0 or k =n then 1;
    elif 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) ;
    fi ;
end proc:
seq( add(add(T(j,k), k=0..n), j=0..n), n=0..30); # G. C. Greubel, Nov 02 2019
  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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[j, k], {k, 0, n}, {j, 0, n}], {n,0,30}] (* G. C. Greubel, Nov 02 2019 *)
  • Sage
    @CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (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(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 02 2019
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.