A173741 T(n,k) = binomial(n,k) + 4 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows.

1, 1, 1, 1, 6, 1, 1, 7, 7, 1, 1, 8, 10, 8, 1, 1, 9, 14, 14, 9, 1, 1, 10, 19, 24, 19, 10, 1, 1, 11, 25, 39, 39, 25, 11, 1, 1, 12, 32, 60, 74, 60, 32, 12, 1, 1, 13, 40, 88, 130, 130, 88, 40, 13, 1, 1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1, 1, 15, 59, 169, 334, 466, 466, 334, 169, 59, 15, 1

Offset

0,5

Comments

For n >= 1, row n sums to 2*A100314(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) + 2*(1 - A103451(n,k)).

T(n,k) = 5*A007318(n,k) - 4*A132044(n,k).

n-th row polynomial is 2*(1 - (-1)^(2^n)) + (1 + x)^n + 4*(x - x^n)/(1 - x).

G.f.: (1 - (1 + x)*y + 5*x*y^2 - 4*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).

E.g.f.: (4 - 4*x + 4*x*exp(y) - 4*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)

Sum_{k=0..n} T(n, k) = 2^n + 4*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Example

Triangle begins:
  1;
  1,  1;
  1,  6,  1;
  1,  7,  7,   1;
  1,  8, 10,   8,   1;
  1,  9, 14,  14,   9,   1;
  1, 10, 19,  24,  19,  10,   1;
  1, 11, 25,  39,  39,  25,  11,   1;
  1, 12, 32,  60,  74,  60,  32,  12,  1;
  1, 13, 40,  88, 130, 130,  88,  40, 13,  1;
  1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1;
  ...

Mathematica

T[n_, m_] = Binomial[n, m] + 4*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) + 4$
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) + 4
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) + 4 >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

Crossrefs

Cf. A100314, A103451, A156050.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), A173740 (q=2), this sequence (q=4), A173742 (q=6).

Sequence in context: A176348 A176264 A195397 * A171147 A171695 A179233

Adjacent sequences: A173738 A173739 A173740 * A173742 A173743 A173744

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