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A132735 Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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: A205117 A077228 A049687 * 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

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Last modified November 30 19:02 EST 2021. Contains 349424 sequences. (Running on oeis4.)