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A173740 T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows. 2
1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 8, 6, 1, 1, 7, 12, 12, 7, 1, 1, 8, 17, 22, 17, 8, 1, 1, 9, 23, 37, 37, 23, 9, 1, 1, 10, 30, 58, 72, 58, 30, 10, 1, 1, 11, 38, 86, 128, 128, 86, 38, 11, 1, 1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1, 1, 13, 57, 167, 332, 464, 464, 332, 167, 57, 13, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..77.

FORMULA

From Franck Maminirina Ramaharo, Dec 08 2018:(Start)

T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).

T(n,k) = 3*A007318(n,k) - 2*A132044(n,k).

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

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

E.g.f.: (2 - 2*x + 2*x*exp(y) - 2*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)

EXAMPLE

Triangle begins:

  1;

  1,  1;

  1,  4,  1;

  1,  5,  5,   1;

  1,  6,  8,   6,   1;

  1,  7, 12,  12,   7,   1;

  1,  8, 17,  22,  17,   8,   1;

  1,  9, 23,  37,  37,  23,   9,   1;

  1, 10, 30,  58,  72,  58,  30,  10,  1;

  1, 11, 38,  86, 128, 128,  86,  38, 11,  1;

  1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1;

  ...

MATHEMATICA

T[n_, m_] = Binomial[n, m] + 2*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) + 2$

create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 08 2018 */

CROSSREFS

Cf. A007318, A103451, A132044, A156050, A173741, A173742.

Sequence in context: A153843 A318795 A099575 * A028275 A173118 A147289

Adjacent sequences:  A173737 A173738 A173739 * A173741 A173742 A173743

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Feb 23 2010

EXTENSIONS

Edited and name clarified by Franck Maminirina Ramaharo, Dec 08 2018

STATUS

approved

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Last modified October 13 23:58 EDT 2019. Contains 327986 sequences. (Running on oeis4.)