login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A173742 Triangle T(n,k) = binomial(n,k) + 6 with T(n,0) = T(n,n) = 1 for n >= 0, read by rows. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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: A109571 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 12 16:18 EDT 2021. Contains 344959 sequences. (Running on oeis4.)