A173742 Triangle T(n,k) = binomial(n,k) + 6 with T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 10, 12, 10, 1, 1, 11, 16, 16, 11, 1, 1, 12, 21, 26, 21, 12, 1, 1, 13, 27, 41, 41, 27, 13, 1, 1, 14, 34, 62, 76, 62, 34, 14, 1, 1, 15, 42, 90, 132, 132, 90, 42, 15, 1, 1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1, 1, 17, 61, 171, 336, 468, 468, 336, 171, 61, 17, 1

Offset

0,5

Comments

For n >= 1, row n sums to A131520(n) + A008586(n).

Links

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

Formula

From Franck Maminirina Ramaharo, Dec 09 2018: (Start)

T(n,k) = A007318(n,k) + 6*(1 - A103451(n,k)).

T(n,k) = 7*A007318(n,k) - 6*A132044(n,k).

n-th row polynomial is 3*(1 - (-1)^(2^n)) + (1 + x)^n + 6*(x - x^n)/(1 - x).

G.f.: (1 - (1 + x)*y + 7*x*y^2 - 6*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).

E.g.f.: (6 - 6*x + 6*x*exp(y) - 6*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)

Sum_{k=0..n} T(n, k) = 2^n + 6*n - 6 + 6*[n=0]. - G. C. Greubel, Feb 13 2021

Example

Triangle begins:
  1;
  1,  1;
  1,  8,  1;
  1,  9,  9,   1;
  1, 10, 12,  10,   1;
  1, 11, 16,  16,  11,   1;
  1, 12, 21,  26,  21,  12,   1;
  1, 13, 27,  41,  41,  27,  13,   1;
  1, 14, 34,  62,  76,  62,  34,  14,  1;
  1, 15, 42,  90, 132, 132,  90,  42, 15,  1;
  1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1;
  ...

Mathematica

T[n_, m_] = Binomial[n, m] + 6*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]

Prog

(Maxima) T(n, k) := if k = 0 or k = n then 1 else binomial(n, k) + 6$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 6
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else Binomial(n, k) +6 >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

Crossrefs

Cf. A007318, A103451, A132044, A156050, A173740, A173741.

Sequence in context: A366494 A133823 A168643 * A146881 A174301 A174378

Adjacent sequences: A173739 A173740 A173741 * A173743 A173744 A173745

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 23 2010

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018

Status

approved