A135226 Triangle A135225 + A007318 - A103451, read by rows.

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 9, 7, 1, 1, 6, 14, 16, 9, 1, 1, 7, 20, 30, 25, 11, 1, 1, 8, 27, 50, 55, 36, 13, 1, 1, 9, 35, 77, 105, 91, 49, 15, 1, 1, 10, 44, 112, 182, 196, 140, 64, 17, 1, 1, 11, 54, 156, 294, 378, 336, 204, 81, 19, 1

Offset: 0

Author

Gary W. Adamson, Nov 23 2007

Keywords

• nonn
• tabl

Comments

Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  5,  1;
  1, 5,  9,  7,  1;
  1, 6, 14, 16,  9,  1;
  1, 7, 20, 30, 25, 11,  1;
  1, 8, 27, 50, 55, 36, 13, 1;
...
		

Links

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

Crossrefs

Cf. A007318, A083329, A103451, A135225.

Programs

GAP

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return ((n+k)/n)*Binomial(n,k);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

Magma

T:= func< n,k | k eq 0 or k eq n select 1 else ((n+k)/n)*Binomial(n,k) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 20 2019

Maple

T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else ((n+k)/n)*binomial(n,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, ((n+k)/n) Binomial[n, k]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

PARI

T(n,k) = if(k==0 || k==n, 1, ((n+k)/n)*binomial(n,k)); \\ G. C. Greubel, Nov 20 2019

Sage

@CachedFunction
def T(n,k):
    if (k==0 or k==n): return 1
    else: return ((n+k)/n)*binomial(n, k)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 20 2019

Formula

T(n,k) = A135225(n,k) + A007318(n,k) - A103451(n,k) as infinite lower triangular matrices.

T(n,k) = ((n+k)/n)*binomial(n,k) with T(n,0) = T(n,n) = 1. - G. C. Greubel, Nov 20 2019

Extensions

Corrected and extended by Philippe Deléham, Nov 14 2011