A111279 -id:A111279 - cp's OEIS frontend

A026736 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 k=(n-2)/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 6, 11, 10, 5, 1, 1, 7, 22, 21, 15, 6, 1, 1, 8, 29, 43, 36, 21, 7, 1, 1, 9, 37, 94, 79, 57, 28, 8, 1, 1, 10, 46, 131, 173, 136, 85, 36, 9, 1, 1, 11, 56, 177, 398, 309, 221, 121, 45, 10, 1, 1, 12, 67, 233, 575, 707, 530, 342, 166, 55, 11, 1

Offset: 0

Clark Kimberling

T(n, k) is the number of paths from (0, 0) to (n-k, k) in 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+2)-to-(i+1, i+3) for i >= 0.

			Triangle begins
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  5,  6,   4,   1;
  1,  6, 11,  10,   5,   1;
  1,  7, 22,  21,  15,   6,   1;
  1,  8, 29,  43,  36,  21,   7,   1;
  1,  9, 37,  94,  79,  57,  28,   8,   1;
  1, 10, 46, 131, 173, 136,  85,  36,   9,   1;
  1, 11, 56, 177, 398, 309, 221, 121,  45,  10,   1;
  1, 12, 67, 233, 575, 707, 530, 342, 166,  55,  11,  1;
  ...

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

Row sums give A026743.

T(2n,n) gives A026737(n) or A111279(n+1).

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n mod 2)=0 and k=Int((n-2)/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, Jul 16 2019

T[, 0] = T[n, n_] = 1; T[n_, k_] := T[n, k] = If[EvenQ[n] && k == (n-2)/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 (* Jean-François Alcover, Jul 22 2018 *)

T(n,k) = if(k==n || k==0, 1, if((n%2)==0 && k==(n-2)/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, Jul 16 2019

def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-2)/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, Jul 16 2019

Offset corrected by Alois P. Heinz, Jul 23 2018

A026737 a(n) = T(2*n,n), T given by A026736.

Original entry on oeis.org

1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841

Offset: 0

Clark Kimberling

nonn

Is this the same sequence as A111279? - Andrew S. Plewe, May 09 2007

Yes, see the Callan reference "A bijection...". - Joerg Arndt, Feb 29 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Callan, A bijection for two sequences in OEIS, arXiv:1602.08347 [math.CO], 2016.

Cf. A026736, A111279, A059714.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-5*x+4*x^2 -(1-5*x)*Sqrt(1-4*x))/(2*x*(1-4*x-x^2)) )); // G. C. Greubel, Jul 16 2019

T[, 0]=T[n, n_]=1; T[n_, k_]:= T[n, k]= If[EvenQ[n] && k==(n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 22 2018 *)
CoefficientList[Series[(1-5x+4x^2 -(1-5x)*Sqrt[1-4x])/(2*x*(1-4x-x^2)), {x, 0, 30}], x] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec((1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2))) \\ G. C. Greubel, Jul 16 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-2)/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, Jul 16 2019

G.f.: (1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2)). - G. C. Greubel, Jul 16 2019
a(n) ~ (47 - 21*sqrt(5)) * (2 + sqrt(5))^(n+2) / (2*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019
G.f. G satisfies 0 = G^2*(x^3 + 4*x^2 - x) + G*(4*x^2 - 5*x + 1) + 4*x - 1. - F. Chapoton, Oct 16 2021

cp's OEIS Frontend

A026736 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 k=(n-2)/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

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