A132735 Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 7, 5, 1, 1, 6, 11, 11, 6, 1, 1, 7, 16, 21, 16, 7, 1, 1, 8, 22, 36, 36, 22, 8, 1, 1, 9, 29, 57, 71, 57, 29, 9, 1, 1, 10, 37, 85, 127, 127, 85, 37, 10, 1, 1, 11, 46, 121, 211, 253, 211, 121, 46, 11, 1, 1, 12, 56, 166, 331, 463, 463, 331, 166, 56, 12, 1

Offset

0,5

Links

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

Formula

T(n, k) = A007318(n,k) + 1 - A103451(n,k), an infinite lower triangular matrix.

T(n,0) = T(n,n) = 1; T(n,k) = C(n,k) + 1 otherwise. - Franklin T. Adams-Watters, Jul 06 2009

Sum_{k=0..n} T(n, k) = 2^n + n - 1 + [n=0] = A132736(n). - G. C. Greubel, Feb 14 2021

Example

First few rows of the triangle are:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  4,  1;
  1, 5,  7,  5,  1;
  1, 6, 11, 11,  6, 1;
  1, 7, 16, 21, 16, 7, 1;
  ...

Mathematica

T[n_, k_]:= If[k==0||k==n, 1, Binomial[n, k] +1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

Prog

(Sage)
def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 1
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else Binomial(n, k) + 1 >;
[T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021

Crossrefs

Cf. A103451, A132736.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), this sequence (q=1), A173740 (q=2), A173741 (q=4), A173742 (q=6).

Sequence in context: A077228 A049687 A362036 * A028262 A173117 A050177

Adjacent sequences: A132732 A132733 A132734 * A132736 A132737 A132738

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 26 2007

Extensions

Corrected and extended by Franklin T. Adams-Watters, Jul 06 2009

Status

approved