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

0,5

Comments

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

Links

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

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

Example

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;
...

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 *)

Prog

(PARI) T(n, k) = if(k==0 || k==n, 1, ((n+k)/n)*binomial(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
(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
(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

Crossrefs

Cf. A007318, A083329, A103451, A135225.

Sequence in context: A296457 A124020 A124234 * A104730 A249488 A275204

Adjacent sequences: A135223 A135224 A135225 * A135227 A135228 A135229

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

Corrected and extended by Philippe Deléham, Nov 14 2011

Status

approved